3. Does Electrum handle all the change/change addresses for me so I don't have to worry about it?
This. When you use a software wallet like electrum then it'll handle change for you and you don't have to worry about it. Change is only a problem for paper wallet users or those who muckabout with private keys directly.
Since you've asked I'll answer your other questions too:
1. Unspent coin goes to a change address. Can I specify this change address?, i.e. create a new wallet and direct the change there? If so, is there an opportunity during a payment transaction to specify the change address I want to use?
The generic term for fine grained control over the inputs and outputs of a transaction is called "coin control". Electrum allows you to determine which inputs form a transaction but does not allow you to select where the change goes. Bitcoin Core allows you to do both.
What electrum does allow you to do is specify whether you want to use separate change addresses, which is good for privacy, or send change back to one of the originating addresses. So if you go to tools > preferences > transactions tab you can select to use change addresses or not.
2. I've setup Electrum on Tails (the wallet lives in permanent storage). Will Electrum remember my wallet/transactions when I power down?
Electrum on tails is an old version so I don't recommend using it. You should use a debian live cd and install the latest electrum on to that. Yes you can do that. The instructions for installing electrum on ubuntu/debian are on the download page of the electrum site.
BTW you've called electrum an online wallet. What do you mean by that? Electrum is not a web wallet or a wallet provided to you by some centralized entity. You control the private keys and they only exist on your own devices.