When should I press garlic and when should I chop it?
That is actually quite controversial in its own way. If you are going to use a garlic press, you should cut the root end off the clove (you can do that a bulb at a time if desired) and give the individual cloves a bit of a crush with the side of a big knife before you press them. If you do that and you have a good garlic press, you can then just pluck the paper from the press to be ready for the next clove. That's fine, if pressed garlic is actually what you want.

Garlic is funny this way. It all depends upon how fussy you want to be. Pressing is about the least "perfect" way to prepare garlic for anything, but it pretty much works for everything. Many people find pressing to be the most convenient way to deal with garlic. If you press garlic you get fresh garlic juice and smudged garlic paste. For almost anything that is adequate. 

If you desire for more than adequate, the kind of superlative awesomeness that Michelin Star judges look for and your little brother would never notice, then you need to break out the knife skills. Slicing, mincing, smearing, crushing, and even pressing give different results, even if those differences are barely apparent to us mere mortals.

I don't think ginger should ever be pressed, and I would not say that pressing is an always adequate technique like I would say for garlic. The best methods for ginger are to grate with a microplane or a ginger grater, to finely dice, to do it in volume with a food processor and then keep in the fridge or freeze in single use aliquots, or to roast and use in big chunks. You might find it helpful to know that you can freeze the whole hand of ginger. Frozen, you can use a microplane and the paper will just drift out of your way.

I'm not aware of anything else for which a press of the garlic press type could be of any use.