1. Yes you are allowed to bid on your own players. HOWEVER, if you win the auction, then that player will be considered a brand new player, so merchandising will be lower. Also, the player will be considered as sold even though you bought the player yourself, and so it will increase the tax on future sales. Also, you will still have to pay the tax on the player you bought. So if you get 80% of the sale back, and you buy the player back for 500k, then you will only get 400k because the other 100k will be taken out as tax.
2. No one can know for sure except the BBs, but my understanding is that it decreases the chance of a shot going in, and shows up as a "Block" when it decreases the chance of a shot going in to 0%. Another way to look at it is as a supplement to defence.
3. I definitely think a PR manager with a specialty is worth it. You can pick up a basic manager with a specialty for a very low price and very low salary, and the PR manager will increase attendance at your arena, and help you win some close games. Definitely worth it.
4. The maximum injury time I have seen is about 5 weeks if I remember correctly, and this can occur irrespective of the level of your doctor, however it is more likely to occur with a lower level doctor. For example, there might be a 10% chance of a 4 week injury with a minimal doctor, and a 2% chance of a 4 week injury with a superior doctor. But, just like in real life, sometimes injuries are going to last a long time irrespective of how good your doctor is.