I'm just going to focus on the first issue. It actually belongs in the Injury forum but someone directed me to this thread so here I am...
Is it possible the issue of chin ups resulting in stiffness is related to slight (or exaggerated depending on your form) extension at the C spine as you near the top of the movement?
My guess is that you do the full (or close to full range) of pull ups and chin ups in which the head has to go past the bar. If this is the case, there must be some extension at along the C spine so you don't hit the top of your head on the bar. And it follows that the T1/2 will also extend.
So IF IF IF the inability to keep a neutral C spine is the culprit to your particular case, you may want to try the following set up. If there are other factors involved (other than the inability to maintain a neutral C spine), this set may only assist in preventing shoulder or elbow injury.
Below is a link to a post I wrote back in 2011. The reason for this set up is that it allows a more natural movement at the shoulder and elbows while performing the pull ups or chin ups.
Now there is a secondary benefit. You can keep a neutral C spine and never have to worry about hitting the top of your head.
When you go this web page, scroll down to the post dated 08-14-2011
EDIT: see the post 08-14-2011 6:05 PM
If you have access to a lat pull down station, the same set up can be performed.
As for doing behind the neck pull ups, this variation places your shoulders in an externally rotated position. Very few can actually do this not end up with shoulder issues down the road. In other words, it's very much debatable if the rewards outweigh the risk on for a large segment of the population.