750 watt inverter fuse?

Yes, the line should be fused or otherwise protected (self-resetting circuit breaker) as close to the battery as practical. The purpose of the fuse here is to protect the wire from getting too hot and starting a fire in the case of a short circuit, rather than primarily protecting the inverter. The inverter probably does have fuses built into it.

You likely don't need anywhere near that large an inverter for a TV and a DVD player, particularly if it's an LCD or LED TV.