Capacitors

Yes. As long as you know what you are doing and you’re aware of the possible risk of changing the frequency balance.
Look at this shoot-off between caps.

And another capacitor test.

Those links are all to film cap comparison. If you have the ability to put in a film cap, it can be much better. However, film caps are significantly larger in physical size and may not fit for the uf value you may need. So, electrolytic capacitors may need to be used in those specific scenarios. Also, this electrolytic capacitor could very well be for a notch filter circuit, which does not affect sound quality as much as other capacitors directly in the signal line.

I usually go for nichicon bipolar caps for speaker crossovers if I need them.

But the OP asked for "other not-polarized." Wouldn't they have said with other "bi-polar electrolytics?"
But good point on the size.

Depends how old, and where.

In series = Safe
Going to ground = Dangerous
Old = High ESR
Newer = Low ESR


If you are replacing a series cap, relatively ( < 10 years) new, you are fine.

If the cap is old, it probably has higher ESR, and that should be taken into account, but in series it's not much of a problem.
The problem is in shunts ( 2nd component in low pass)  where loosing a couple of ohms can dump the speaker impedance to 2 Ohms, and cause you to loose amplifier output.