I've seen the EM field tensor mostly written with 2 upper indices, $F^{\mu\nu}$. Does this imply that it's necessarily a (2,0)-tensor (that is, it has 2 contravariant components and 0 covariant ones)? If so/if not, why?
Just as a note, I'm a 3rd year physics student, so my knowledge of tensors is limited compared to that of a grad student.
Every quantity that you are used to, in physics, are usually defined in terms of the upper indices. The nice property of this is that, because upper indices correspond to vector components, this means that if you want to visualise any upper index in terms of vector directions, this will automatically work.
However, that makes a lot of mathematical handling awkward. This is because, after studying enough mathematics, you will realise that the organisation of which differential operators match with which physical quantity, will only make sense if a few things have lower indices. In the case involved here, the treatment of the electromagnetic Faraday tensor will only match up with the mathematical theory of differential forms if the Faraday tensor is naturally having lower indices. That is, Faraday tensor is naturally a (0,2)-tensor because the relations $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ will only match up with $F=\mathrm dA$ if it is a (0,2)-tensor and not the upper indices form.
This will be particularly obvious if you stop using Cartesian coördinates.
Fundamentally, as is the case with gauge fields in general, it's a second degree differential form, $F = ½ F_{μν} dx^μ ∧ dx^ν$, so that its components are those of an anti-symmetric rank two covariant tensor, $F_{μν} = -F_{νμ}$.
For non-Abelian gauge fields, there is also an upper index associated with an underlying Lie algebra basis $(Y_a)$, with $F = F^a Y_a$ and $F_{μν} = F^a_{μν} Y_a$. It is related to the potential one form $A = A^a Y_a = A_μ dx^μ$, whose components are $A^a = A^a_μ dx^μ$. The electromagnetic field (pre electroweak days) is subsumed by this, having a hidden Lie index $F^γ_{μν}$ for a 1D Lie algebra whose sole basis element would be $Y_γ$. In post-electroweak days, it is a part of non-Abelian gauge field (the electroweak field) and the index $γ$ and associated basis element are one of 4! Instead, it's the hypercharge field that has Maxwell-like equations, not the electromagnetic field; while the field-potential relations for the electromagnetic field - in the context of electroweak theory - are non-linear, and part of those for the electroweak field, itself.
The relation between the potential and field are given, component-wise, by $$F^c_{μν} = ∂_μ A^c_ν - ∂_ν A^c_μ + f^c_{ab} A^a_μ A^b_ν,$$ where the structure coefficients $f^c_{ab}$ are those that arise from the Lie brackets $$[Y_a, Y_b] = f^c_{ab} Y_c.$$
For Abelian gauge fields, the Lie bracket is trivial $[u,v] = 0$, and all the $f^c_{ab} = 0$, so the expression would reduce to: $$F^c_{μν} = ∂_μ A^c_ν - ∂_ν A^c_μ.$$ In the language of differential forms $F = dA$.
For the non-Abelian case, you can extend this to $F = dA + A^2$, provided you adopt the convention of setting $[Y_a, Y_b] = Y_a Y_b - Y_b Y_a$ (i.e. by embedding the Lie algebra into an "enveloping algebra") and allowing the $Y$'s and $dx$'s to freely intersperse.
Adopting the indexing $x^0 = t$, with Cartesian coordinates $\left(x^1, x^2, x^3\right) = (x, y, z)$, the potential one form has components $$φ = -A_0,\quad = \left(A_1, A_2, A_3\right),$$ respectively for the "scalar" (or "electric" potential) $φ$ and "vector" (or "magnetic" potential) $$. Adopting the convention $d = (dx, dy, dz)$, the potential one-form could be written as $$A = ·d - φ dt.$$
For the field strength, the components $F_{μν}$ correspond to the $(,)$ fields, with $$ = \left(F_{23}, F_{31}, F_{12}\right),\quad = \left(F_{10}, F_{20}, F_{30}\right).$$ After noting that $∂_0 = ∂/∂t$ and $\left(∂_1, ∂_2, ∂_3\right) = ∇$, you can write the field-potential relations as $$ = ∇×,\quad = -∇φ - \frac{∂}{∂t}.$$ Adopting the conventions $$d ∧ dt = (dx ∧ dt, dy ∧ dt, dz ∧ dt),\quad d = (dy ∧ dz, dz ∧ dx, dx ∧ dy),$$ the corresponding 2-form can be written as $F = ·d + ·d ∧ dt$.
For non-Abelian fields, $(φ,,,)$ generalize to $\left(φ^a, ^a, ^a, ^a\right)$, and the respective field-potential relations would be given by $$^c = ∇×^c + ½ f^c_{ab} ^a×^b,\quad ^c = -∇φ^c - \frac{∂^c}{∂t} + f^c_{ab} φ^a ^b.$$ Adopting the same "enveloping algebra" representation as before, this could be written as $$ = ∇× + ×,\quad = -∇φ - \frac{∂}{∂t} + φ - φ.$$
You may see the expressions $F^∇_{μν} = ∇_μA_ν - ∇_νA_μ$ in the context of curved (Riemannian) space-time geometries. Though it's equivalent to the formula $F^∇_{μν} = ∂_μA_ν - ∂_νA_μ$, so that $F^∇ = F$, the two expressions are no longer equivalent when going over to Riemann-Cartan geometry (which is the geometry you need to use, when fermions are included in the dynamics). In that sense, $F$ is not really an anti-symmetric order 2 tensor - at least not one you operate on with covariant derivatives - but an order 2 differential form. Instead, it is $F^∇$ that is.
Curiously, I've never seen any context where the difference between $F^∇$ and $F$ has been called out. It does become an issue when you're in Riemann-Cartan geometries. (Even when the torsion is zero, you still have a difference between the two off-shell!) There is an empirical question to answer there: which one is the electromagnetic (or more generally, the gauge) field: $F^∇$ or $F$?
In the contexts where you see the contravariant expressions $F^{μν}$, usually, it's in reference to the response fields - which correspond to $(,)$. If you re-insert the (hidden) Lie index, it's actually in a lower position, $(_a, _a)$. In a field theory derived from an action $S = \int d^4 x$ associated with a Lagrangian density $$, the response fields are the respective derivatives: $$_a = \frac{∂}{∂^a},\quad _a = -\frac{∂}{∂^a}.$$ They're not components of tensors, but actually of tensor densities - as should already be clear from the contexts where they're used (e.g. flux density for $_a$). The response field tensor density $$_a^{μν} = -\frac{∂}{∂F^a_{μν}},$$ has components $_a = \left(_a^{01},_a^{02},_a^{03}\right)$, and $_a = \left(_a^{23}, _a^{31}, _a^{12}\right)$.
In the cases where the Lagrangian density is quadratic in the field strength that leads to the Maxwell-Lorentz action $$S = \int d^4 x,\quad = -¼ k g^{μρ}g^{νσ}\sqrt{|g|} F_{μν} F_{ρσ},$$ and more generally to the Yang-Mills action $$S = \int d^4 x,\quad = -¼ k_{ab} g^{μρ}g^{νσ}\sqrt{|g|} F^a_{μν} F^b_{ρσ},$$ the corresponding response fields are $$^{μν} = k g^{μρ}g^{νσ}\sqrt{|g|} F_{ρσ} = k \sqrt{|g|} F^{μν},$$ and for the non-Abelian case $$_a^{μν} = k_{ab} g^{μρ}g^{νσ}\sqrt{|g|} F^b_{ρσ} = k_{ab} \sqrt{|g|} F^{bμν},$$ using the metric $g_{μν}$ and its inverse $g^{μν}$ to lower and raise space-time indices.
For the components $g_{μν} = \text{diag}(-c^2,1,1,1)$, this works out to the respective cases $$ = \frac{k}{c} ,\quad = k c ,$$ for the electromagnetic field, and $$_a = \frac{k_{ab}}c ^a,\quad _a = k_{ab} c ^a,$$ for the non-Abelian gauge field. In the electro-magnetic case, this corresponds to the constitutive laws of the Maxwell-Lorentz field: $$ = ε_0 ,\quad = μ_0 ,$$ with $$k = ε_0 c,\quad \frac1k = μ_0 c.$$ For non-Abelian case, it generalizes to the constitutive laws of the Yang-Mills field $$_a = ε_{ab} ^b,\quad ^a = μ^{ab} _b,$$ with $$k_{ab} = ε_{ab} c,\quad k^{ab} = μ^{ab} c.$$
The permittivity $ε$ and permeability $μ$ effectively generalize to the gauge group metric $k_{ab}$ and its inverse $k^{ab}$, respectively. For Yang-Mills fields, there is the added prescription postulated that the gauge group metric is adjoint-invariant which - for our purposes - means that it makes the structure coefficients totally anti-symmetric: $$f_{abc} = -f_{bac},\quad f_{abc} = k_{ad} f^d_{bc},$$ (where $f^d_{bc} = -f^d_{cb}$, and thus $f_{abc} = -f_{acb}$ is already given by a fundamental property of Lie algebras, i.e. $[u,v] = -[v,u]$).
Normally, in the Physics literature, the metric $k$ is trivialized to Kronecker deltas by choosing the Lie basis $(Y_a)$ appropriately. Only products of semi-simple and Abelian Lie groups have invertible adjoint-invariant gauge group metrics. So, the additional condition on the metric limits the scope of Yang-Mills fields to these types of Lie groups.
In flat Minkowski geometry where the metric $g$ is constant, and the coordinates can be re-indexed (e.g. with $x^0 = c t$) to make $\sqrt{|g|} = 1$, thereby allowing you to conflate tensors and tensor densities, so that you can use $F^{μν}$ tensor more or less interchangeably with the response field tensor density $^{μν}$.
It seems best to me to interpret the Faraday tensor $\mathbf F$ as an operator, mapping vectors to vectors. The input is the 4-velocity vector $\mathbf u$ of a test particle, and the output is (proportional to) the Lorentz force acting on that particle. Given the standard way to write vector components is with a raised index, the most convenient placement of indices for an operator on vectors is then $F^{\mu}_{\hphantom\mu\nu}$. The Lorentz force is $ma^\mu = qF^{\mu}_{\hphantom\mu\nu}u^\nu$, where $m$ is the mass of the particle, $q$ its charge, and $\mathbf a$ its 4-acceleration. We could also write the RHS as $q\,\mathbf F\llcorner\mathbf u$ or $q\,\mathbf F\cdot\mathbf u$ (readers should not trust my signs).
In the language of differential forms, $\mathbf F = d\mathbf A$, where $\mathbf A$ is a 4-vector potential. The $d$ here is called the exterior derivative, and this equation is normally understood as sending a 1-form (field) to a 2-form. However, I prefer to redefine it as sending a vector to a bivector. (The metric allows this conversion.) Then think of $d$ as giving the "4D curl" (my terminology) of the vector field. [In the standard introductory definition, curl returns a vector field. However this only works in 3D. It generalises to arbitrary dimension if we interpret curl as outputting a 2-form instead. However I personally prefer a bivector because the intuition is simpler and more direct.]
Apologies that some of this is beyond the 3rd year level. Indeed for anyone, certain nuances are not the most conventional way to think of things. But I hope at least some aspects are helpful and understandable. If anyone wants resources on this perspective, my biggest influence is geometric algebra.
A tensor of type $(p, q)$ is $p$ contravariant and $q$ covariant. Taking bases, this implies it has $p$ upper indices and $q$ lower indices.
When an inner product is available then we can change the type, that is we can raise and lower the indices. It's natural to think that despite this they still describe the same object.
All this goes through for tensor fields on a manifold where the inner product is generalised to a metric.
Thus we ought to say $F^{\alpha\beta}, F^{\alpha}{}_{\beta}, F_{\alpha}{}^{\beta}, F_{\alpha\beta}$ all describe the same tensor field $F$.
