NB Your first question is improperly stated as Qmechanic pointed out in his comment. I interpret it in a precise sense: If there is a reason why $\Phi$ is supposed to depend at most linearly on the first derivatives of Lagrangian coordinates.
I guess you are considering generalized Lagrangians of the form
$$L(t,q, \dot{q})= T(t,q, \dot{q}) - \Phi(t,q, \dot{q})\:, \tag{-1}$$
for classical systems described in a generalized coordinate system and also taking holonomous ideal constraints into accounts if any. In this case the kinetic energy $T$ takes the form
$$T(t,q, \dot{q}) = \sum_{i,j=1}^n A(t,q)_{ij} \dot{q}_i\dot{q}_j + \sum_{j=1}^n B(t,q)_j\dot{q}_j + C(t,q)\:. \tag{0}$$
It turns out that the matrix $A(t,q) = [ A(t,q)_{ij}]_{i,j=1,\ldots, n}$ is symmetric an positively defined and in particular is invertible. Suppose that
$$\Phi= \Phi(t,q)$$
If you write down the E-L equations,
$$\frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j}=0\:, \quad \frac{dq_j}{dt} = \dot{q}_j\:, \quad j=1,\ldots, n \tag{1}$$
using the fact that $A$ is invertible you see, with a tedious computation, that it is possible to re-write these equations into the precise form
$$\frac{d^2q_j}{dt^2} = F_j(t,q, \frac{dq}{dt}) \quad j=1,\ldots,n\:.\tag{2}$$
where in particular, for some functions $G_k$ we have
$$F_j(t,q, \frac{dq}{dt}) = \sum_{k=1}^nA(t,q)^{-1}_{jk} G_k(t,q, \frac{dq}{dt})\:. \tag{3}$$
The form (2) of Euler-Lagrange's equations is said to be normal. This is a general notion in the theory of ordinary differential equation systems of order $n$ and just means that
the derivatives of highest order $n$ can be separated, and inserted in the left-hand side, from the derivatives of other orders $n-1, n-2,\ldots, 0$ which appear in the right-hand side in any functional form.
If the right-hand side is sufficiently regular (jointly continuous and locally Lipschitz in the variables $(q, \dot{q})$), the existence and uniqueness theorem establishes that any system of 2nd-order differential equations of the normal form (1) admits a unique (local and global) solution as soon as you fix the state of the system at initial time:
$$q(t_0) = Q\quad \dot{q}(t_0) = \dot{Q}\:.$$
This property is the mathematical translation of the determinism principle of classical physics.
The crucial facts to pass from (1) to (2) are that (a) the first time-derivatives $\dot{q}_j$ appear quadratically in (0), (b) they do not appear in $\Phi$ and (c)
that $A$ in (0) in invertible.
The same result can be obtained if $\Phi$ is also function of the $\dot{q}_j$, but they appear therein linearly.
Any different dependence, in particular a quadratic dependence of $\dot{q}_j$ in $\Phi$ could give rise to an obstruction to reach the normal form of the Euler-Lagrange equations, so that the principle of physical determinism may fail to be satisfied.
It is worth stressing that linearity in $\dot{q}_j$ appearing in $\Phi$ is only a sufficient condition to fulfill standard hypotheses for the existence and uniqueness theorem. So one may construct physical systems respecting the determinism principle, but described with Lagrangians including potentials with non-linear dependence on $\dot{q}$.
However the only two cases of generalized potentials $\Phi(t,q, \dot{q})$ known in classical
physics, the potential of electromagnetic (Lorentz) forces and the potential of general inertial forces respect this linearity constraint.
Inserting time derivatives of order greater that $1$ in $\Phi$ gives rise to the same type of problems regarding the determinism principle, though these higher order derivatives are not completely forbidden and they are used in some semi-classical models, to describe the self-acceleration of an electric charge in particular.
