18
The rules don't specify that
Athletics. {snip}
• You attempt to climb a sheer or slippery cliff, avoid hazards while scaling a wall, or cling to a surface while something is trying to knock you off. (Basic Rules p. 62)
The PC rolls an ability check when the DM calls for one.
Ability Checks
An ability check tests a character’s or monster’...
16
1. Is it possible to exceed your normal maximum jump length by passing a Strength (Athletics) check?
This is a bit situational. Say, for example, a PC only has a Strength Score of 5. If there is a 10 ft gap, and if the player cannot make it, it's death. If we use the rule for a long jump, the PC is going to die.
But, this rule about a long jump is what the ...
13
Yes
You've done all the math and that's correct. Of course, you need to be using a skill check in which you're proficient, but you are a 20th level PC and built it in such a way to maximize the skill check possibilities.
7
Is an artificer's homunculus servant's passive Perception affected by its Might of the Master trait?
Yes. Passive Perception is based on Perception modifier.
The rules on passive checks state:
Here's how to determine a character's total for a passive check:
10 + all modifiers that normally apply to the check
If the character has advantage on the check, add 5. For disadvantage, subtract 5. The game refers to a passive check total as a score.
...
4
It's up to the DM.
Since there are rules for jumping as well as a spell which modifies that ability called jump, the DM may very well resort to using the rules given for jumping on page 182 of the PHB.
However, the DM may call for an ability check when an action has a chance of failure, or when the outcome is uncertain. I think latter applies here. ...
4
Depends on your stats and proficiency, but best case is about 10-25%
For the most part, the difficulty is based on the level of the monster using this chart. As you can see, it is anywhere from level + 14 to level +25. This can be further modified by creature rarity, with uncommon creatures adding +2, and rare adding +4 or 5, and so on.
So, in order to ...
1
The closest would probably be a tool proficiency - stonemason's tools might be an appropriate proficiency for most of the examples you give. The last one, though, I'd have be a History check.
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