There's an Achievement box on the profile but it won't show any achievements. Can someone assist me by explaining how I get some of my supposedly 96 achievements to show up?

One bonus question, I have trained my pets in Health and their max health goes up but current health does not... do I need them in the 'resort' to gain full health maybe?

The achievements widget in the profiles hasn't worked since the achievements system got a revamp several years ago. So for now there's no way of making it work. To increase your current health you can visit the healer

That's sad... for years it hasn't worked? :(

I think the new achievement system came about in 2014 or 2015, so yeah it's been a few years at least since that happened. I don't remember if achievements on profiles were working before then, but I don't think they've worked since.