I need help understanding how particles do what they do and maintain the structures they maintain if so many of them exist for such a short time?
In the case of the nucleus and pions, pions only exist for mere nanoseconds. So if they decay, and their properties change, what is holding that nucleus together?
Are the nucleons continually exchanging a stream of pions that perpetuates the interaction that holds the nucleus together?
Like I said... Explain like I'm five.
Electromagnetism has a field Maxwell knew of. What he didn't knows is it's quantized: it can be observed as not only waves but also an integer (but not fractional) number of photons. You can do something similar to observe pions from a quantized field for the residual strong potential between nucleons.
Observable pions decay quickly. Virtual pions are something I need to speak of with care; as is often noted on Physics SE, "virtual particles" don't mean what non-experts often think they do. The most mundane characterization of them is as a mathematical fiction that rephrases certain facts. For example, the resulting potential is of the form $-\frac{g^2}{r}e^{-mcr/\hbar}$, on a one-mass-$m$-species-exchanged approximation.
It's not that a pion is made, exchanged before it quickly decays, and then another pion is made; it's that the field responsible for the nucleon-nucleon potential can be used to create observable pions, but if it is they'll be short-lived.
Your misunderstanding is an amalgamation of multiple misconceptions perpetuated in science education media.
1: So yes, perturbative calculations involve the exchange of virtual particles, but they're just approximations to what's really happening: the field configuration follows the Feynman path integral formulation, which we can't calculate analytically, nor really visualize.
2: The vacuum doesn't change. It's invariant under space and time translations, and between one observation and the next, it takes all possible paths (per (1)). There is an old fashioned way to calculate that where particles pop in and out of existence, violating energy conservation per the uncertainty principle, but that was superseded in 1948 by relativistic quantum field theory. The vacuum just is, and it's not nothing.
Note that particles popping in and out of existence is clearly not invariant under boosts, e.g: what is the average momentum? Which was does it point? Is it zero in all frames? it's just not a relativistic way of thinking: it has to be frame dependent.
3: A stable nucleus is the ground state of a many body quantum system. It is an energy eigenstate, and thus a stationary state: it does not change. For instance: all alpha particle are identical, everywhere, and always. Nevertheless, you can use the effective field theory of meson exchange to calculate properties. It does not mean they are a boiling froth pion exchange in real life. Likewise for 3b, though atoms are simpler since we can treat them perturbatively.
3: The nucleons in a nucleus do not have distinguishable proton and neutron identities. For instance, a deuteron has the hadronic content:
$$ \frac 1 {\sqrt 2} \big(|pn\rangle - |np\rangle\big) $$
So, each nucleon is not in an isospin ($I_3=\pm \frac 1 2$) eigenstate, but together they form an eigenstate of total $I_3=0$. The math is isomorphic to angular momentum, which is why it's called isospin .
So, in summary: don't take virtual particles too seriously, and don't take classical descriptions of quantum systems too seriously. They are both helpful in understanding, but can generate more misunderstanding if taken literally.
The 5-year-old inside me has since the age of five always thought of this in the following way. (If this way of visualizing the unvisualizable is bogus, please let me know and I'll stop using it immediately.)
A pion consists of two quarks. Imagine having a proton and a neutron right next to one another, as in a nucleus. Each consists of three quarks.
It is possible for a quark inside the proton to experience a brief close encounter with a quark inside the neighboring neutron. For that fleeting moment, you could draw a circle around those two quarks and declare the temporary pairing to constitute a pion.
Since both those quarks remain bound inside their respective nucleons, they will still be interacting with the other quarks inside those nucleons and in this sense, their transient pairing is a means by which the two nucleons can interact.
Then the two quarks continue their dance and the pairing is no more, and the thing we called a pion is gone.
