I am using the program "Bitcoin Core" to store my coins, but I was wondering how to collect the new "Super Bitcoin"? I suppose "Bitcoin Core" will not support this new crypto, so I am looking for a good software wallet.
The Super Bitcoin developers should announce which wallets or exchanges will support their coin. Which ever you use, remember to have access to your private keys. I assume (but it's only an assumption) that Bitcoin Core would allow anyone to claim the forked coin because this is the original Bitcoin wallet.
Do installing another wallet requires to re-download the whole blockchain once again?
No.
You only download the blockchain when you install the Bitcoin Core. You're essentially running a node, you're part of the blockchain network, when you work with Bitcoin Core.