will I be able to do that without the help of an Electrum server?
Yes.
For example will I be able to export my private key to a web wallet for using my bitcoins there? Or any other possible way?
Exactly that way.
You'd use Wallet -> Private Keys -> Export (then decrypt your wallet, if you've encrypted it)
You'll get a list of all private keys currently included in your loaded wallet file and the according addresses.
You then can export those either by hand or as csv/json file.