In reply to charliesdad:
As others have said it sounds like you're doing a lot. Do you keep a training diary? And if so are you seeing regular strength gains? If so then what you're doing is probably OK. If not you might be doing too much, not giving your body enough time to recover.
The most specific climbing exercises are things like campusing and finger tip pull ups. For the former you obviously need a campus board. However for finger tip pull ups all you need is a bit of wood and way of fixing it to whatever you intend to fix it to. For instance if working out at a gym you could hang it or bolt it to a pull up bar or other structure.
Ordinary pull ups are good too, especially if you're not wanting to work your fingers on a certain day. To make them more specific to climbing (and harder) do them with your knees raised or in an L-hang position. This will work your core too.
Not sure why you're doing pull downs if you're also doing pull ups too. They're a very similar movement and pull ups are the better exercise. So I'd consider ditching those.
The core is important for climbing so you definitely want to include that. Planks type exercises are good for climbing, particularly one-legged type of exercises. Make sure you keep good form. You can alternate plank exercises with crunch based movements and bar movements knees to elbows or toes to bar type stuff. Choose the exercises to suit your fitness level.
Many climbers rate deadlifts. I think they depend on how your back is. Good all round strength builder but they will drain you if you do too much. Kettlebell swings work similar muscle groups and are less hard on the back and possibly more specific to climbing.
For finger strength you could do heavy finger curls with dumbells. I've seen some comp climbers recommend these.
Finally doing some flexibility stuff will help climbing. The very successful Japanese team do 2 hours of stretching every day. Yoga is a good way to this. Watching experienced Yoga practitioners climb is always a joy and inspiring to see.
Post edited at 18:38