Can Under 12s and Under 13s head the ball, and what are the training limits?

In matches: yes. Under 12 is the transition point where heading comes back into the game, as players move to secondary school. The deliberate-heading ban (FA Rule O) only covers Under 7 to Under 11.

In training, The FA's heading guidance for Under 12 and Under 13 says heading remains a very low priority:

• No more than one session a week that includes heading.
• No more than 5 headers per player in that session.
• Use light balls.
• Deliveries should be self-served or over short distances.
• Heading practice should always be unopposed.
• Focus on introducing correct technique.

For comparison, from Under 14 to Under 16 the limit rises to one session a week with up to 10 headers per player.