"Also, when are the distinctive features not supposed to matter? How to stop the players from misusing them?"
Players tend not to abuse it in my experience, particularly as you need to be spending Hope for Inspired to provide a mechanical bonus, so it's not like every roll ends up bogged down by a debate over whether a distinctive feature applies.
This guidance from the rulebook is good: "A trait can be invoked on a roll only if, based on its description, it is reasonably plausible for someone with that quality to fare better than an individual without it."
If your players can make a good case for why they're better at it due to the distinctive feature, then go for it!
Distinctive featurs are used a lot in my experience as both a player and loremaster. Both because they make a big mechanical difference (2d6 makes a big difference to succeeding and getting 6s for extra successes) but also because they frequently help a character shine at what a player wants their character to be good at. If you've taken Swift, then your character performing great feats of Athletics matters to you!
- Keen-eyed helping on important Awareness/Scan rolls.
- Patient boosting a key Craft roll for repairs.
- True-hearted being deployed with various social skills to help in negotations, as the other character was convinced by the speaker's sincerity.
- Leadership (Captain distinctive feature) boosting Battle rolls to gain advantage.
- Fierce to boost Awe rolls to Intimidate Foes.
"What skills and situations are affected by e.g Eager, Patient or Honourable?"
- Anything a player can reasonably justify why they're better at it due to the distinctive feature, based on the distinctive feature's description.
- Eager might affect a social roll when your enthusiasm helps, but not help with some other NPCs. "You are filled with excitement and impatience when an endeavour piques your interest." Or a Craft roll where appropriate (something that piques your interest) or your excitement might boost an Athletics check.
- Honourable: your belief in doing what is right will spur you on if giving a speech at a tribal judgement for someone you believe is innocent, but not when you have good reason to think they're guilty (but support them for political/financial reasons etc). It could inspire an Enhearten roll in a social setting or to Rally Comrades in battle if you feel your cause is honourable.
- Patient: again, can help social rolls in the right circumstances, or with the right audience. Or a craft check to do a methodical repair. Or perhaps your Battle plan is based on patience, or your patient approach helps in your hunting by waiting for just the right circumstances. Patient might not help with a Stealth check when you suddenly hear an orc patrol coming, but will help if you're preparing an ambush.
Hope this all helps!