what can I sell and what can I keep: beerstein, luxury pipe, fancy chessboard and other treasures?

The only purpose of treasures is to sell them to make money. Some can be combined with others, so be sure to examine them first before you sell them.

Examine it- if it says something like: it has 3 crevices, wonder what there for? or has 2 divots, maybe something could fit inside...
anything like that means keep it for now, if not, sell it how it is

No point of keeping treasures. The whole purpose of them, is to sell them to the merchents for money. You can combine some treasures for more money. The treasure with the highest price is the Salazar Family Crown. selling 42,000 PST. You can get that by combining the crown, the crown with jewel and the royal insingnia together.

The treasures are only there for you to sell. Here's how to know what to sell, and what to keep (so you can combine items with that treasure to increase its value):

The treasures that have "divots" on them (it'll tell you if it has them when you examine the treasure in your menu), are to keep. There are various items, such as Catseyes (for the Beerstein) and the Gems (for the Masks) which are combined to those two items to make them more expensive.

The treasures that don't have divots are for selling, such as Spinels, Antique Pipe, Velvet Blues, and Amber Ring.

The only Treasures worth keeping are...

Red, Blue, Green Eye
Butterfly Lamp

Red, Green, Yellow Catseye
Beerstein

Red, Purple, Green Gem
Elegant Mask

Blue Stone of Treason, Green stone of Judgement, Red Stone of Faith
Golden Lynx

Crown Jewel, Royal Insignia
Crown which is later renamed Salazar Family Crown

All of these listed are combination pieces that go together and drastically increase each pieces value. Once each piece is connected to each other, they're no longer worth keeping.

Beerstein: keep for putting more gems in
luxury pipe: sell
fancy chessboard: sell

some can be combined for more money