I was thinking about this while playing yesterday.
Jumping should definitely not use stamina, specifically for the reason of getting stuck. There were a few times I definitely got myself stuck in areas that I wouldn't have gotten out of if not for jumping (like the time I nearly stranded myself on the island you have to build a bridge to before I put the second bridge head down).
I do think gathering should use stamina - but maybe only at first. Perhaps as you level your gathering up the stamina used goes down? So at level 5 gathering it goes down to 1 stamina, at level 10 gathering you no longer use stamina.
Running using stamina makes sense; do you wanna get to the mines earlier in the day but have less stamina to spend, or vice versa?
It's come up in another thread here, but I think the argument boils down to: should stamina regen? I don't think it should, at least not without some fancy ring crafted at a high-level work station using rare components, but as came up in said other thread, we definitely need more stamina regen items, and early-game ones (such as apples) need to be easier to come by without expending more stamina to get them than they actually regenerate.