BackGammon
Public Domain - no problems
Ludo
Not sure, but it dates back to the late 19th century. It's derived from Pachisi, which dates back to the 6th century.
If not sure, give it another name
Reversi
Reversi is a trademark of Ravensburger AG. Just give it another name, no copyright on the game mechanics
http://www.mpatric.com/2011-05-01-reversi-is-a-trade-mark
Chess
Public Domain - no problems
Draughts / Checkers
Public Domain - no problems
Mahjong
Public Domain - no problems
Poker / BlackJack / Solitaire
Public Domain - no problems
Dominoes
Dates back to like 13th century - no problems
Connect 4
Public Domain - no problems
Here's an extensive list of Public Domain Board Games:
https://www.boardgamegeek.com/boardgamepublisher/171/public-domain
In general you can say that you can copy all of gameplay without any issues at all. That is not copyrightable or enforceable. The things that matter are assets - art, sound, music, video, etc.
An easy-to-implement AI you could use for chess and draughts is called the alpha/beta algorithm with beta cutoff.
I used it once for a Connect 4 AI Competition. Was not the fastest when it exceeded search depths >8, but it sure did beat all other AI's in the competition. It smoked them big time, that is
Nice project, btw. I'm a sucker for board games. If you need some beta testing, drop me an eMail.
P.S. I think you need to change the font. Already lost one eye trying to decipher it
Cheers,
PSY