Q: After losing your ship to destruction does the insurance cost cover exact replicas of all engineered modules or do you end up with modules minus any mods?
Yes. The insurance will cover every engineered (and non engineered) module exactly as is at the time of destruction. No engineered modification penalty is incurred whatsoever. They are exact replicas.
Q: Does the insurance cost of your ship change at all due to adding mods?
Negative. Engineering modules has absolutely zero effect on the insurance cost/re-buy of your ship. The only things that dictates the price of re-buy is the cost of your ship and the cost of your components, which is why it's generally a smart idea to get things from price-discounted systems before modding them. Discounts range from 10 to 20% and a ship discount coupled with a few module discounts can significantly lower your re-buy cost.
The post above me answered thoroughly enough everywhere but this next question, so this is the actual reason for my post....
Q: Does removing or storing engineered modules affect them in any way? I.e. mods get removed?
The term "removing a module" could cause confusion here.
Currently in the game, you have the ability to STORE your modules. This effectively "saves" your module in the current station you are storing it in the exact current condition of said module. If it is engineered, you lose absolutely no attributes whatsoever. It doesn't even cost you any credits to put a module into or remove a module out of storage.
After you store a module, you can have it shipped to various locations for a fee and a time delay.
Besides storing your module, you can also do a few other things to it:
One such thing is the option to SELL the module back to the station. This is done at no credit loss to the player, but if the module is engineered, it does not gain any monetary value and the modification is lost upon selling the module (you can still buy it back for a short period of time post sale).
Another thing you can do to your modules is actually REMOVING THE MODIFICATION, which is where the confusion might stem from. This option strips the engineered modification from the module and calls it a day. The module stays in your ship, but it's restored to its default, per-engineered values as if you just got it from the station. This action cannot be undone.
You also have the option to SWAP a module, but this is used to simply move a module from one optional internal/utility mount/hardpoint to another compatible similar slot.
Cheers.
