Rule for several items instead of one rule for each

There are several answers to your question:

  1. Yes, you can run rules on group members. A good place to start is here:
  1. You can set multiple when statements for a rule to trigger the same rule with different items. See the use of the implicit variable triggeringItem here:
  1. Your best bet, in this case is probably to just do this without rules at all. Item profiles can set timestamps for you automatically: