TOO MANY QUESTIONS!
In the past decade, the world has seen a kind of renaissance of board games. The world has long moved past Monopoly and Stratego and standard entertainment fare. There are literally hundreds of new games coming out every year for every type of person in every conceivable genre. It would take me hours to even begin answering your questions, there’s just so many factors that go into it.
I’ve seen the frum oilem only partially embrace it (Ticket to Ride and Settlers of Catan are somewhat popular in heimishe homes) which is surprising considering that the content of most of these games is rather benign by moral standards and very few have any issues in regards to Shabbos. What annoys me the most is the frum publishing companies that make their own games just slap a few extra themes (“reskins”) on boring old existing titles like Candy Land when there’s so much potential for so much more. Seriously people, look up your local tabletop game store (yes, such a thing exists) and ask for a recommendation. Barring that, head out to Barnes and Noble and ask them.
I like games that are short (30-60 minutes), don’t involve too much luck, and can be played with 2-4 people. Right now my family is really enjoying Forbidden Island which is a co-operative game where everyone has to work together to collect treasures on an island before the water rises too high.
I have a few others on my “I want to play” list. There’s Hanabi, a co-operative game where everyone can see each others cards, but not their own, and have to tell the other players what they have in their hands, but the rules prevent giving away too much information.
Then there’s Sabotuer, which works in large groups. Everyone works together to dig for the treasure, except for one or two people who have to prevent the others from finding it. And no one knows who they are, so it often devolves into a shouting match of accusations and counter accusations.