Let's clear a general misconception: Bitcoin that are on an address of which you do not own the keys are not your Bitcoin. That being said, keeping a lot of money of online services and exchanges is definitely discouraged. There's a good chance that you will get burned.
Bitcoin core for full wallet. Electrum for SPV wallet.
Indeed.
Bet we'll see some replies recommending exchange wallets.
Unfortunately, people with limited knowledge tend to give out bad advice.