Batteries operate by the principle of ion formation. The chemical process is exothermic with respect to the transfer of electrons from the cathode to the anode. So the more charge that is moved the hotter the batteries get. This is often the limiting factor for charge conversion.

Capacitors on the other hand simply store charge using electrostatic field attraction, the only barrier to their current flow are the i2r heat generated in the conductive paths (a superconducting supercapacitor for example could dump all of its charge instantly without any problems, if such a thing existed, its manufacturer would be worth more than Apple :-)

Flywheels store energy mechanically as angular momentum, they have good energy density, and can return it quickly by being attached to a generator, but are generally hard to deal with in systems with an external acceleration because their tendency to precess if that acceleration results in a rotation that is perpendicular to the flywheel. The proposed carrier ones I've seen are shown as being on gimbals for that reason.

No, I expect they'd need to be specially designed for it, with a number of compromises (gas venting/replenishment) along the way.

Battery technologies tend to have a limiting internal resistance that determines the maximum discharge rate, although in many cases the maximum SAFE discharge rate is much lower - traditional lead acid will allow you draw so much current that the plates buckle and the acid boils...

It's all down to the chemistry, and every technology has different characteristics and weaknesses - for example, any secondary cell involving nickel or zinc has to deal with the tendency of these metals to grow dendrites when plating out of solution (ie. the recharge case) - this is what bursts a regular AA cell if you try to recharge it using DC. There are simple workarounds, and although the regular dry cell design is not optimal for recharging it can be done. It also causes the memory effect in NiCd cells, and explains why they can sometimes be recovered with a large pulse of charging current (local melting of the dendrites).