What's 'best' is going to be largely a matter of personal preference, and as such I would recommend getting some samples if you're unsure. The site you've linked to offer sample photo books and print colour tests on different papers for a small fee:
This will be the best way to check you're getting what you want.
As for some general, less objective, advice, I've worked at a commercial printer and have experience with different photo labs for my own prints, so my two cents.
I find glossy papers tend to oversaturate colours. Sometimes this is
the effect you want to achieve, but for portraiture or skin tones (as
in weddings) this tends to go over the top. I think a lustre finish
or matte finish offer more natural skin tones, and this is what I
prefer. I would suggest the metallic paper might be too glossy for this reason, but you would need to get a sample to be sure!
Glossy papers also tend to show fingerprints and smudges more, so I
also avoid them for this reason for prints that will be handled, as
in a book. A printed photo to frame is another matter...
Gloss feels 'tacky' to the touch, whereas matte feels more luxurious
and smooth. Lustre is somewhere in between, but more than adequate
With this in mind, if it were me choosing a book I would opt for lustre, matte as a second choice, and I wouldn't bother with gloss. But that's just my preference!
As for the paper thickness, this isn't a measure of 'quality' of the paper used, which tends to be measured in gram(me)s per square metre (or 'gsm'), but literally how rigid you want the papers to be.
2mm would be like a child's card book (in terms of rigidity at least!). 0.8mm is closest to A4/legal thickness. 1.3mm is somewhere inbetween.
Although I wansn't able to confirm this from the artisanstate website, I would imagine the 'quality' of the paper, regardless of the thickness ordered, will be the same. The images are being printed on photo quality paper, which is typically around ~200 - 300gsm, which is a good quality paper (A4/legal tends to be ~80-90gsm).
I hope all this helps, but the papers and finishes look very good on the website, so I would imagine you'll be happy with whatever you order! If this answers your question, don't forget to mark your question answered so others know this is closed!