Will a sector colonize planets in Stellaris

Looks like, eventually... yes!

Couldn't tell ya what triggers it, though.

Additionally, I found this line in the 00_strategic_resources script file:

influence = {
    accumulative = yes
    AI_category = influence
    max = 1000
    sector_ignore_cost = yes
}

Which seems to explain how Sectors handle the influence cost (They ignore it! Those cheaters!)

And, confirmed by Paradox Developer: https://twitter.com/Martin_Anward/status/737198287254654976

Sectors do not have influence, they ignore influence costs when building. Outposts cost for country in Clarke.

Well, random is random, so yes - despite your best efforts, sometimes you get stuck with a sub-par hand. But there are a few ways to help turn lemons into lemonade.

First, if you find yourself getting boxed in by other empires early, there are a few 'new game' settings you can tweak to make things a little smoother. Experiment with any combination of the three until you find settings you are happy with:

• Galaxy Size
  • A bigger galaxy with the same number of player means that everyone tends to be more spread out. A possible downside is it may make for longer games (or an upside, depending on your point of view).
• AI Empires
  • Lowering the number of players also means everyone is generally more spread out. And the requirements to win (% of planets owned) haven't gone up.
• Advanced AI Starts
  • Lowering this means fewer AI player get a head-start, which means fewer enemy borders get inflated unnaturally fast.

Second are some general tips that may help you keep pace:

Explore fast and explore early. The single science ship the game starts you with is nice, but unless you're on a particularly tiny galaxy map, you need to cover a lot of ground; I typically start by building a second science ship and recruiting a leader for it IMMEDIATELY (If you luck out and can recruit a leader with survey speed+, go for that one). Use the map (M) and hold Shift while right clicking on a series of systems to queue up a bunch of surveys so that they can do it largely on auto. Start close to your homeworld and work your way out.

While all this is happening, you want to research colony ships, and build up your infrastructure so that you can build one ASAP. But hold off on actually building it until your science ships have encountered at least one colonizable world, because the upkeep on colony ships is insane. Once the ship is built, if you have multiple suitable worlds surveyed; go for the furthest one. You can probably pick up the closer one later, where as the further one stands a greater chance of being swallowed by another empire's border by the time you get a second colony ship.

Be stingy with frontier outposts, those things cost a lot of influence, and each one you have active increases the time it takes to recoup that loss (and when all your leaders start dying of old age, you NEED that influence). I, personally, try to avoid using them unless there's a particularly juicy grouping of energy/mineral rich systems (multiple instances of 6+ of one or both), or a habitable planet that's a little too close to another empire's border to risk waiting. In the late game, yes - you'll want to be using them a little more strategically to push back other borders to access a colonizable world, or to keep a border from expanding over a key system that gives resources you can't afford to lose.

But in the end, there are only so many star systems, and eventually they are ALL going to fall under someones' borders. Once this is true, you have a few options; try to ally your way to victory, or take them by force. To expand (a bit) at this point without fighting, you need to have researched colonization of other planet types, and possibly terraforming. This will allow you to colonize planets within your borders you otherwise couldn't. As those populations grow - if they were near your borders they may help push your borders outwards, exposing more systems... up to a point.