If I could get a ceramic capacitor at the capacitance of 10uF and within my voltage requirements, which from my initial searches I can, what problems would I experience if I were to change, if any?
Some circuits (like some linear regulators, for example), require a certain minimum ESR from the output capacitor, which could cause the circuit to oscillate when using a ceramic but not with an electrolytic.
In a precision circuit, a ceramic might not be preferred due to microphonics, but in those cases you probably wouldn't want an electrolytic either.
Otherwise, ceramics are generally preferred. They'll have lower ESR, they're not polarized, they need less voltage de-rating, and so on.
Finally, when searching SMD footprint standards, the common packages seem to be 0402, 0603, and 0805, where they increase in physical size respectively, but also power rating, which suggests to me I should use as large of a package as possible
Usually you choose the smallest package you can get away with because you want to fit as much circuit as you can in the smallest footprint.
Also, for ceramics, the larger sizes (1210 and higher) can have reliability issues because they can be cracked if the board flexes.
In an aluminum electrolytic capacitor, the positive terminal is connected to the foil that has the oxide layer on it, and the negative terminal is connected to the one without the oxide layer. This puts the negative terminal in direct contact with the electrolyte, and the case (assuming it has no insulating liner) is also in contact with the electrolyte.
Therefore, if you measure the resistance between the case and either lead, the negative lead will have relatively lower resistance to the case in both directions, while the positive lead will show essentially infinite resistance in at least one direction.
If the case is insulated, you can try applying a small bias voltage (3-5V) to the capacitor in each direction (through a current-limiting resistor of 100K or so) and see which direction allows the least current; this will be the correct polarity of the capacitor. This works because an electrolytic capacitor has a weak diode action as well. For further details, see the Wikipedia article.
Best Answer
For a low-pass filter for PWM where the voltage is always the same polarity yes, you can use multiple electrolytics:
simulate this circuit – Schematic created using CircuitLab
You might also consider a "Pi" RC circuit:
simulate this circuit
For lower losses it is better to replace the resistor with a suitable inductor to make an LC circuit.