Do NHL teams save any cap space with LTIR? If so, how much of the player's contract?

No, when a player is on LTIR the cap hit remains counting against the team no different than if they were healthy.

Then what's LTIR for?

LTIR can be used to exceed the salary cap.

While they are exceeding the cap they are not accruing cap space like they would be if they were below the upper limit.

I'm still confused ... isn't that basically giving the team cap space?

Yes, teams do receive "relief" from layers on LTIR. Look at what CHI did last season w/ Marian Hossa or this season w/ DET w/ Zetterberg. DET's cap hit is currently over $87M-well exceeding the upper limit of the NHL salary cap.

So do teams get to save the whole cap hit or only some of it?

DET currently has just over $10M available via LTIR. They can use all of that $10M as long as their current team cap hit is under the upper limit of the cap ($79.5M). If or When Johan Franzen returns from LTIR DET would need to clear out whatever excess salaries they have in order to activate him. That's my interpretation of the rule.

But if a team has a player with a 5M cap hit and they are 4.5M over the cap, could they put the player on LTIR and not make any other moves? Would that be allowed? Or could they only do it if they were over the cap by less?

You know an entire write up can be found here:

https://www.capfriendly.com/ltir-faq

answers all of your questions

Just look at DET this season. Look at their team page on this site-forget cbs.com or ESPN or any of that. They'd be non-compliant if they were to activate Zetterberg and/or Franzen of LTIR. To answer your question directly...In theory I suppose they could try to circumvent the cap rules to their advantage, say if one of their more expensive players was struggling, but I've never heard of a healthy player being played on LTIR just to get over on the rules. That would be very devious.

I hope I was able to summarize this a little bit. The full page that was referenced on this site can be a little overwhelming if you're not completely familiar w/ the process. I, myself haven't read the entire page, but like I said, look at how both CHI and DET managed their respective cap situations and that should provide a better understanding.

**COUGH[Maple Leafs]COUGH**

If they knew they'd have a better chance at Austin Matthews any team in their position would do the same. And look at them now...they almost have too much talent up front lol.

I'm not sure it helped their chances at Auston Matthews, but it likely sped up their rebuild/turnaround.

When a player is cleared to play by the doctors the team has to make the room for them to be activated

It isn't up to the Leafs (or team) to determine if a player is eligible for LTIR, it up to the league. As we all should know, Horton was a LTIR, hence the trade with Columbus, and the Leafs spend 27.5m out of pocket to take on that LTIR.
Lupul is different situation, but still it the league and Lupul who decided he wasn't NHL fit, not the Leafs.