Is it survival?

Not at the moment, but there will be some survival mechanics.

Will I lose the ship I've paid real money for if it gets destroyed?

Only if you let your insurance lapse.

How does insurance work

https://robertsspaceindustries.com/comm-link/transmission/12820-Insurance-FAQ-And-Update

how much is it, how long can I have it for?

Depends, it will be purchased with in game $ only.

I simply refuse to play if I'm paying $80+ for a ship that gets lost forever if I lose it.

It's a possibility in this game only if you make some very, very stupid decisions.

If I have no insurance and I lose it will it cost a lot to get back?

Don't know, but it depends on the ship. In game ship prices aren't known yet.

Which package should I buy?

One of the starter packages, nothing more. The Mustang is more nimble, the Aurora is tough, and more utility oriented.

What is there to do in the game right now?

It's listed in the sidebar.

My personal recommendation: don't get it until alpha 3.0 releases, and even then, only after you do a bit more research on the game on your own.

This is a crowdfunded game actively being developed, and there is always a chance the game can flop, so any money you spend should be money you can afford to lose forever in a worst case scenario.