If I have nothing else to do in the sector, and there is a large gap between my ship and the Rebels position, do I get any advantage in the next sector?
No. In fact, rushing through sectors places you at a disadvantage.
Exiting the sector early does not give you more time in the next sector. The rebels start in the same position, begin advancing at the normal time, and advance at the same rate they normally would (unless you're in a nebula system).
Speeding through a sector and exiting it as soon as possible reduces the amount of resources you will gain per sector. This will make the rest of the game much harder. To increase your chances of success, it's recommended that you rush toward the exit beacon, but then explore as many new systems near it as you safely can before the rebels catch up. You'll enter each new zone with more scrap and gear, which will dramatically increase your chances of success.
In my experience, it's better to explore the sector if you have time because you'll probably need the scrap in later sectors to buy better weapons, ship upgrades, and repairs.
However, if you are heavily damaged or running out of fuel, and you don't think there are any nearby shops, it can be beneficial to quickly go the next sector to avoid taking more damage or fuel loss. It's true that you aren't guaranteed to find a shop quickly there either, but it should be your top priority to find repairs rather than risking fights that could potentially end your journey.
According to the FTL wiki, the number of stores per sector is:
- Start: 1-2
- Civilian: 2-3
- Engi: 2-3
- Zoltan: 2
- Mantis: 1-2
- Pirate/Rebel: 1-2
- Rock: 2
- Abandoned: 2
- Nebula: 0-1
- Slug: 0-1 (with 2 nebula stores)
- Crystal: 2-3
Shaun is generally correct. The only thing that I would add is that you may want to rush through a sector if you're in a bad situation. Some possibilities are
- Heavily damaged and there are no stores nearby to repair at
- Running low on fuel (or missiles if you rely on them)
- In a sector that you're not well prepared for (since many sectors have similar types of enemies, and they can be a bad matchup for the equipment you have)
So you should generally try to get as many encounters as you can, but it is good to know when you should lay low for a bit and try to get yourself to a better situation before resuming the fight.