50GB will most likely take weeks to see any noticeable coin accumulation if you are mining on a pool.
I will try to say this simply. Wallet are just an interface to the blockchain network. Local wallet has this software interface installed on your computer. Online wallet has this interface installed on a remote server. They both access the same thing, but people prefer their own computer for security reasons.
Star wallet and stop wallet. I am assuming you see this on Qbundle. Qbundle manages the Burst wallet software to make it easy to use for Windows users. You can start the wallet or stop it from there, yes. If it is “stopped”, it is not connected to the network.
To run a local wallet (on your computer), yes you have to sync all of the blockchain database with the network. Most people choose to mine with a pool, in which you don’t need to have to run a wallet on your computer.