How does the capsule keep the pilot from getting injured by the extreme acceleration the ships are capable of?
The pilot is housed in a tank of ectoplasmic fluid - “pod goo” - that keeps the body safe from inertia effects and sorts out all the biological stuff.
^ There is that, but also civilizations of EVE are capable of gravity manipulation (see: Mining lasers, tractor beams), it is likely they use this or some other technology to negate inertia entirely, else the crews would be human slushies every time you warp.
I don’t think this technology has been explained in detail in any of the official lore material.
If they have warp drive and jump gates, I am certain there is some way to offset inertia. If there were not, no other ship would be any better off.