My hunch is that the part is 100nF, and your replacement is 500nF.
I wouldn't risk using a cap without a voltage rating indication on it. Also, you're better off replacing that Z5U dielectric with one that's more temperature stable, like NP0.
I'd source a 100nF, 2kV NP0 capacitor as a replacement.
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.
Best Answer
From the datasheet.
Most capacitors don't actually have a "current" rating, since that doesn't make much sense. You can't put a sustained current thru a capacitor anyway. If you tried, its voltage would rise linearly, and then you'd get to the voltage limit where you'd have to stop. Put another way, current thru a capacitor is inherently AC.
Capacitors do often have a ripple current spec. Capacitors designed to be used in applications where this matters, like switching power supplies, will have a ripple current spec. Check out the Panasonic FK series, for example. These are designed for particularly low ESR (for electrolytic capacitors). Applications where low ESR is important are likely to subject the capacitor to significant ripple current, so this is specified.
Other capacitors are optimized for different purposes where the maximum allowed ripple current isn't relevant, or is so bad the manufacturer doesn't want to admit what it is. In cases where you care, don't use such capacitors.