Wallets and exchanges are two different things, If you want to store your coins securely then I suggest using Electrum for Windows and Mycelium for Android. These wallets allow receiving and sending clearly so you will be able to deposit into exchanges, do what you have to do and then withdraw later.
There is no wallets that could make change your BTC to USD and keep it there, If you want a wallet that have built-in exchange (for other coins) then Exodus.io would be the best choice.
And i'm guessing that Electrum and Mycelium would be synced? or are they separate accounts?
So if you can't exchange to USD permanently then selling your coins to other people is the only option to get USD? if so, where would be the best place to do so?
I know of a company that will do this in the future but who knows how long it will take to get banks to sign off on it, might be years from now.