So I want to start using Electrum. Do I use it to access my Armory wallet or do I have to create a new wallet with Electrum and make a transaction from Armory to it?
No. The wallet formats are incompatible. You'll either need to:
a. Export your private keys from Armory and import them into Electrum
b. Export your private keys from Armory and "sweep" them into Electrum
c. Send all your coins from Armory to Electrum
I would recommend C. (Note that b. is effectively the same thing as c. but more time consuming and complicated)
Also, if I (for some reason) lost my wallet but know the password, could I retrieve it or would it be gone? I noticed the app creates a file that stores the wallet locally. So, if that file is gone forever does it mean my satoshis are gone forever too, or is there a way to get my satoshis back in case that file went missing?
When you create a standard wallet in Electrum, it will give you a 12 word "seed mnemonic". This works in the same way as the "Root Key" paper backup does in Armory. It enables you to completely restore the wallet regardless of whether or not the file has been deleted, your computer exploded or you forgot the wallet password.
electrum seed is not bip39 so you can't use the coleman site to generate address specific private keys.
You can modify 2 lines of code in the iancoleman website and have it generate the keys from an Electrum seed mnemonic