I am afraid that I may have lost my bitcoins or that I fucked up. Could someone explain what is going on and if I lost my bitcoins?
You did not lose your bitcoins.
Armory just doesn't have a connection to bitcoin core (since you don't have it installed yet). Therefore it doesn't show your the balance (it has no database to lookup).
All you ever need to access your bitcoins are the private keys (make sure you have a backup!).
It says that it is 4 years behind and it should take around 5 days to complete synchronizing.
If you don't want to completely download and sync bitcoin core, you can always download a lightweight client (e.g.
electrum).
Such a SPV client does not need to download/process the whole blockchain. It is relying on a server to get the necessary information about your wallet (balance/transactions/..).
How is it that I check my balance in Blockchair? Do I select bitcoin explorer?
A blockchain explorer is simply a website which has the whole blockchain stored and indexed.
You can just visit the site (e.g.
blockchair.com or
blockchain.info) and enter your address or transaction id.
If it shows 'confirmed', everything is fine and you have received your coins properly.