Is chess Turing-complete? Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff the program halts?

The rules are the same as ordinary chess minus the 50 move rule, exchanges and castling.

And what is the minimum number of different types of pieces (i.e. the simplest game) that is needed for a chess-like game to be Turing-complete?
(Each type of piece having a set of allowed moves that is invariant under translations)
The classic reference on this is the paper "Computing a Perfect Strategy for nxn Chess Requires Time Exponential in n" by Aviezri Fraenkel and David Lichtenstein; unfortunately, I can't find the (30-year-old!) paper online anywhere and while I think I have the proceedings it came from, it's buried in a pile of boxes.  Of course, your board isn't finite, but your initial piece configuration is; this means that if there's any computable bound on the needed board size to resolve an initial position, then chess isn't Turing-complete.  (If the status of all positions of diameter $d$ could be determined using boards of size no more than $f(d)$, then they can be determined in $\mathrm{EXP}(f(d))$ time by the EXPTIME completeness result, and thus wouldn't be Turing-complete).  I vaguely recall seeing something about such a result, but unfortunately some quick searches aren't turning anything up.  The expert on the mathematics of chess is Noam Elkies, so hopefully he can spot this and chime in...
As chess is played on a finite board with finite many pieces, the number of which can not grow but decline during the game, chess is a game with finite many states and thus never Touring-complete. Though the number of states grows exponentally with the size of the board chess only represents a finite state machine and will be won by computers at some time in the future.