Do NHL players invited for a tryout sign a tryout contract?
Do they pay their own way?
What if they get hurt in camp, are they covered by the team?
The only contracts under which you can play a game in the NHL regular season are the NHL Standard Players Contract (SPC - Exhibit 1 of the CBA) and a one day Amateur Try Out Agreement (ATO - Exhibit 17 of the CBA). The NHL ATO is used only for emergency conditions and does not pay any compensation.
Players in camp may also sign an AHL SPC, ATO, or PTO with the clubs minor league affiliate. The PTO (Professional Try Out) contract is a minor league contract of up to 25 games, but can be terminated earlier. I believe players on PTOs would earn at least the AHL minimum salary for the duration of the contract (LS may have a better answer here) and would be covered by the teams insurance..
What is a PTO in the AHL?
It seems that there is no place on the internet that properly explains the AHL PTO, so I will do my best to fill that void. Please let me know if there are any errors.
PTO = Professional try-out contract
This is an animal found only in the AHL. PTO's are for 25 games, but the player can be released before that. He can also be signed to a second PTO, but after two PTO's he either becomes a free agent or is signed to a SPC - standard players' contract (also just called an "AHL contract").
Another type of contract in the AHL: ATO = Amateur try-out contract
These are signed by previously amateur players -- graduating college seniors and players finishing juniors. There is no set number of games on these.
Any player on an AHL SPC, PTO or ATO must be released from his contract before he can sign in the NHL. Occasionally you'll hear people say an NHL team "picked up his contract," but he's really signing a different kind of contract, it's not directly transferable. The player is not eligible to play in the NHL on a PTO or even a SPC. To play in the NHL you must be signed to an NHL contract, which are full-year, guaranteed contracts (with exceptions for goaltenders in emergencies and players sent back to junior. There is also a one-game ATO in the NHL which is extremely rare, but used for players coming out of college sometimes. See Justin Abdelkader of Detroit).
The AHL CBA is only available for a fee from the PHPA (the AHL Players Association) website and does not seem to available on the AHL site - so I can't check if PTOs are subject to any minimum salary requirements.