1. Answered in full above.

2. More examples needed, but from the words you presented there it seems like your dictionary does this for some unknown reason as I can easily give you the plural short form of хороший. Хороши.

3. Russian adjectives form three sub-groups. I'm not sure what the exact translations of those groups would be in English, so I'd just use their Russian names:

Качественные прилагательные - they describe a quality that can be a varying attribute of a noun, changing to a varying degree. E.g. Быстрый - Some people can be very fast, some will be even faster than that, some will be just fast enough to be destinguished as fast but no more.

Относительные прилагательные - they describe a quality that cannot vary in any way. E.g. Правительственный or зубной for that matter. Someone or something can be one of those, but they can't be more/less dental, more governmental, etc.

Притяжательные прилагательные - these are just possesive adjectives, e.g. мамин, папин etc.

Of all those groups only качественные прилагательные have a short form.