I can't tell offhand. Your shifter won't have any problems, but your derailer might not be able to handle the extra chain length. And you should go for a new chain. Chain and sprockets wear together, and may well get upset if you replace only the one. On top of that it's rather important to have enough chain to survive the big-big combo, as pretty bad failures can happen otherwise.

A 7-speed freewheel is wider than a 5-speed. At least it will require respacing the wheel. It'll probably be a tight fit in the frame as well.

Have you considered swapping out the crankset instead? I was looking at an Altus double just the other day, which would have suited your purpose.