English question - please help?

You need to do more exercise.
You need to do more exercises.

These are both correct.

You need to do more exercise. is more along the lines of 'get off your butt and do something', while 'You need to do more exercises.' would be more appropriate if someone is exercising and they need to do a wider variety of exercises.

Dogs can swim would be the most correct option. Dog can swim would need to be converted to either 'That dog can swim' or 'A dog can swim' to be correct.

Jeremiah just has the right answer... Dogs can swim is right.
Dog can swim would need an article, that is A dog, for instance, it should be:
A dog can swim, This dog can swim, that dog can swim.

The correct ones should be

You need to exercise more/You need to do more exercise

Dogs can swim