This is a *superb* can opener for £1.79 as I paid £8.99 for something very similar several years ago.
The best way to remove the lid is with a fingernail. As you say, the small jaws on the side of the can opener are very fiddly to use as they are small and difficult to see without bending down.
I don't think you need worry about the lining of the can deteriorating when exposed to oxygen. That might have been a worry when tin was used as the internal coating; I don't think that tin is in regular use any more.
There is no need to clean the tin opener by immersing it in water. It only needs the large cutting roller to be wiped, if that.