Are these addresses just for one time use or can I use them normally just like I use my Legacy address?
You can reuse segwit addresses an infinite amount of times, just like legacy addresses, but please, do not put your private key on that website or on any website ever, ideally never export your private key, your wallet software should never show your private key. To move your founds from one wallet to another, then just send yourself the coins from one address to another.
Not so sure about the Electrum part as I'm still not getting it, but all I can see is that, addresses are getting changed back to back whenever I get out of some apps (GreenBits and GreenAddress) and get back in. If it will be like this, how will I be able to receive coins multiple times over a single address when there will be a change after each transaction? Why can't we see a private key instead, so that we might just put it in some compatible wallet and start using it?