The NHL is not getting rid of the cap. And with the league being US-based, it won’t cowtow to the EU. It’ll just give an ultimatum & move on.
But that's the assumption here. Canada joins the EU. EU says "you can't have a draft / salary cap". NHL says "nu-uh" - which causes the Canadian teams to have to leave the NHL.
It most certainly would. Canada is such a cash cow that the NHL never expands into it. Ever notice that? New revenue lines in further US expansion would occur. Houston, Atlanta, Phoenix, Austin, San Diego, etc., would take their place. There would be short term pain, but over time, the league would be just fine.
And who do you think Montreal & Toronto would play? They would be the only ones who could afford to keep playing in a non-capped league. Maybe Vancouver, but questionable. Edmonton drives revenue only due to McDavid. By the time we are talking about, he’ll be long retired. Calgary & Winnipeg would have no shot. I suppose Montreal/Toronto could join one of the Euro leagues, but does that make sense? Even the NFL is hesitant about overseas expansion.
Edmonton drives revenue from a lengthy playoff run, but mostly due to having a nearly brand-new arena that they're experts at squeezing every drop out of. Edmonton is not a richer city than Vancouver of Calgary - Calgary in particular if they get a new arena will shoot up in revenues.
Canada absolutely is a cash cow for the NHL - but it already has all the best markets sewn up. Winnipeg and Ottawa are already second tier teams (financially). While I'm sure you could put a team in QC, or Hamilton, or Saskatoon or wherever and they'd do ok they wouldn't be driving league revenues like Toronto, Edmonton or Montreal do.
Let’s talk talent too. While salaries may be hypothetically higher in Canada, it’s all relative. High end players routinely favor US teams over Canadian ones. Taxes for certain, but better opportunities in the US as well. The money is in the US.
Not always true, but to the extent players prefer US markets it's due to A: taxes B: weather (Buffalo doesn't get any love) and C: less media attention.
Here’s something else to consider too. The NHL would retain ownership of its intellectual property. Perhaps even the names of Montreal & Toronto or enough of it to make both names/history moot. Most certainly the Stanley Cup.
That's the easiest one to answer. Go search the US Patent and Trademark office online search.
tmsearch.uspto.gov
You'l find the TM for the Toronto Maple Leafs is owned by: MLSE. Interesting I can't find a registration for "Montreal Canadiens", but just "Canadiens" is owned by CLUB DE HOCKEY CANADIEN, INC. - the Canadiens holding company. And so on and so forth.
So go ahead, join the EU. You’ll lose hockey over it.
Maybe not lose it - but it seems like you'd get two separate leagues, with very different business models - and both much poorer than they were compared to the old NHL. The CNHL would be dominated by a few big clubs, with the second tier celebrating when they have a "good enough" season, never expecting to win the championship.
The USNHL on the other hand would continue with the same business model - but without some of its biggest most noticeable clubs risks becoming irrelevant in the US sporting marketplace.
But again - this is just a fun "what if" - because it'll never happen.