I am considering transfering my bitcoins from coinbase to electrum wallet. My question is, what happens if Electrum suddenly shuts down and their servers go offline? Will I be able to access my coins using another wallet or will they be lost?
I use electrum myself, and as a user of it I can tell you that your coins are safe there. If and only if it went down your bitcoin can easily be recovered as long as you've taken the precaution of writing down your recovery key which is generated when you make your wallet. The development team behind electrum is also pretty awesome, so I doubt they would shut down without notice.
It looks like only expectations and believes. Each one should save keys on your offline storage.