Far-fetched means unlikely. If you say that someone's idea or proof is far-fetched, you are saying that it is unlikely to be correct.
Far afield means distant or wide-ranging. This doesn't quite fit either, at least to my American ear. I expect far afield to describe distance traveled ("I've gone far afield of my home") or a difference between two topics ("I started in biology but now my studies have gone far afield of that, into literature and mathematics").
Perhaps the best adjective I know for this is roundabout, which is an antonym of direct:
circuitous, indirect < had to take a roundabout route >
I might say:
This is quite a roundabout solution; maybe you can think of something a bit shorter? (or: ...something more direct?)
To "bite the bullet" means to accept that something unpleasant or difficult is sure to happen. It is often used when someone has tried easier or simpler alternatives for a task, but those methods have not been successful; however, trying other alternatives before going to the harder way is not required.
An easy way to remember this definition is to imagine a soldier biting a bullet while having emergency surgery on the battlefield with no medicine for pain. (This is often considered the origin of the phrase, but there is little evidence to support that claim.)
On the other hand, to "grit one's teeth" means to prepare for a challenge or strengthen one's will. This is a more general term, which can be used for a person dealing with any challenge, not necessarily an unavoidable one.
Anyone who is "biting the bullet" will probably "grit their teeth," but just because someone is "gritting their teeth" it doesn't necessarily mean they've "bitten a bullet."
EDIT to add requested examples:
Bob's car broke down on a highway in the middle of the desert. He tried to fix it himself but didn't know how. He tried flagging down someone passing by for a ride or to borrow a phone but nobody stopped. After a couple hours, Bob decided to bite the bullet and walk the 25 miles back to the last service station he passed.
Alice really wanted to make her school's football team. She was already tired from a day at the gym but she gritted her teeth and told herself to do one more set of push-ups so she could beat the competition.
Best Answer
"One's" is one of the "most correct" methods of expressing general possession.
"Your" is more modern, but is considered by a few to be grammatically incorrect because it specifically refers to "you" when you might want to refer to someone in general, and can have a "slang" feeling to it because of this. But, because of it's extremely frequent use, "your" has generally been adopted as being correct.
Either method is valid, and what you use depends on how you want to come across. "Proper" or "modern".
The same applies to the usages of "one" and "you", with "he or she" and "they" being similar replacements.