This looks fun, even though it seems only tangentially related to the thread. (As an aside, it seems to me this is a more useful exercise if you're planning on running 13th Age or The Pool than any flavour of D&D, though it might be somewhat useful for Next or for 1E with secondary skills.)
In good sailing conditions, yes.[*]I would be automatically able to sail a small boat without needing to roll dice
Yes[*]I would know how to tie common knots
All else being equal, yes.[*]I would be better at tying knots than someone who is not a sailor
All else being equal, no.[*]I would be less good at tying knots than someone who has high dexterity
No.[*]I would be able to shoot and hit an unaware person at 50' with a bow without needing a roll
Yes.[*]I would need to make a roll to shoot someone who was aware of me
Virtually all game systems, even "one-roll" ones, differentiate between accuracy and damage in some way. I'd lean toward giving you more of the latter, but not necessarily any more of the former.[*]I would be more effective with my bow than a weaker person because I can pull more weight
Depends what you mean by "dexterous". In D&D this is a very broad category that runs together a lot of things - manual dexterity, general grace and agility, and accuracy - that are quite different. In a system that defines dexterity less broadly, it might not help your archery at all. In D&D, it does so almost by definition, with the mirror image of the above caveat.[*]I would be less effective with my bow than a more dexterous person
Yes.[*]I would be better at shooting my bow on a ship than a non-sailor
Yes, though whatever benefit you're getting from your strength probably accounts for this already without attempting to handle it explicitly.[*]I would be better at shooting my bow on a ship in a storm than a weaker sailor
Maybe.[*]I would be worse at shooting my bow than a "soldier" with "good skill at archery"
No, since that's only a very, very small part of being a bowyer.[*]I would be awesome at repairing bows because not only am I good at archery, but I know how to tie knots and handle rope
You don't generally repair those, you replace them. Almost any attempted repair will just snap the next time you try to use it. But you'd be pretty good at stringing a bow since you have the most important thing you need - strength - and certainly know enough about bows not to damage it in the process. (Again, the sailor aspect is contributing almost nothing here.) Stringing a bow is non-trivial - consider how big a deal being able to string Odysseus' bow was.[*]I would never need to roll to repair a basic bowstring, given the materials I need
Not without needing a roll, but that roll should be very heavily in your favour.[*]I would always beat someone at archery if they were "bad at archery" without needing a roll
Only if they had some way (e.g. a past history with you) of knowing that, or the setting was such that this is legally enforced in some way.[*]I would always get a better rate of pay for sailing a boat as a "sailor" than someone who was "highly charismatic" but not a "sailor"
If the difference is large, sure. Even if it's close, the mechanics used should be such that you're heavily favoured (d20 is really bad at this).[*]I will always beat someone in a tug-of-war game if they are less strong than me
No. Tug of war doesn't require that much skill.[*]Since sailors are very used to playing tug-of-war, I'd expect to beat someone equally strong as me if they were not a "sailor"
If you mean on sight, no (no I don't agree with the statement, not no you wouldn't need to roll). You might pick it up from enough or the right kind of interaction with them, though not necessarily if they were actively trying to hide the fact.[*]I do not need to roll dice to tell if someone else is a sailor.
Again, it depends what you've seen of them. You certainly can't tell this just from a casual glance at someone doing something non-archery-related.[*]I do not need to roll dice to assess someone's skill as an archer
Same as above.[*]I do not need to roll dice to assess how strong someone is