Are there penalties?
Anything else I should know?
The rules don't cover every possibe situation but I would say they still need at least one hand to climb and if that hand is being used for climbing it can'twield a weapon.
But can you use armor spikes while using a one handed weapon and climbing?
*Runs away
Sarcasm aside, I would have to say no, unless you/your GM can justify that you only need your feet to climb in a given situation.
As Howie said. even with the spider climb spell you still need your hands free. That said, if your climb speed was from or like the slippers of spider climb. this would allow you to have your hands free as per the items description.
If you stea... er, borrow from v3.5, there was a skill trick in Complete Scoundrel called "Corner Perch". With sufficient skill ranks in climb and a suitable corner to brace yourself in, you could get both hands free. But if you screw up the check, you can fall off.
From RAW, probably not easily. You could give them potions of Levitate or Fly and have them just hover above and fire arrows.
From a house rule perspective? I'd allow it (have allowed it) with Spider Climb. The spell says that you have to have your hands free in order to climb; if you're not moving, you're not climbing (i.e., making use of your climb movement speed) - you're just standing there (even if it's standing 30 feet up on a wall). If you've used a weapon in that round at least one of your hands is occupied, and you can't climb anywhere else that round (so you can move then shoot, but not vice versa).