How and should one start up a Mopar club?

Yes, I've done it. The club still exists although due to my life outside of cars I'm not active. I do still have the "Founder" membership card that was never supposed to expire...lol.
If you want to use the logo and names in any advertising you have to have written permission from the corporate owner of the logo and names. Back in the early 90s it was from Chrysler, and I had to write this person explaining what the club was for and how the copyrighted items would be used. It was non-profit which makes a big difference. I must have done it right because the club still exists, it's never had a legal issue with Chrysler, and has a good and active membership. Due in large part to the head monarch I think. I hope to rejoin when I have time for it.
The alternative is to just start a club that's not brand affiliated and follow your local laws for non-profits. Otherwise at any time the corporate lawyers can descend on you and shut it down.