Stolt förälder

It's better to use 'stolt matte' (female owner) or 'stolt husse' (male owner). Using 'förälder' for a pet is seen as a bit of a joke.

Yes, in the context of parenting, it is almost always positive. Only in very specific contexts of arrogance does it become negative.

The plural is 'stolta föräldrar'. Note the 'a' on 'stolta' and the 'ar' on 'föräldrar'.

You say 'Jag är stolt över dig.'

Many people just say 'Stolt mamma' or 'Stolt pappa'.

Only if you are talking about your personal life. It's not a professional term for business results.

'Mallig' is a more childish word for 'proud' and often implies showing off in a silly way.

Yes, 'förälder' can refer to any parent regardless of gender.

You say 'stoltaste föräldern'.

Yes, it's a standard way to describe parental characters in novels and plays.

Yes, that is very common and perfectly correct.

There isn't a single phrase, but one might say 'en besviken förälder' (a disappointed parent).

No, but it comes from a word that meant 'stately' or 'standing tall'.

Yes, but keep it 'lagom' (moderate). Don't overdo it!