Yes, you can use the cheap meters. They probably won't have an audible feature to allow you to 'beep' a connection... but just use the ohm feature and when you have good solid connection the meter will read '0' as opposed to 'OL' when the circuit is open.
The parts in the tank could be anything... I'd just dump them out. If the sender is reading within range I wouldn't worry about it. If it's not there isn't much you can do about it other than replace it.