How long can you leave it?

Sorry if this is in the wrong thread. I am pretty new here. My question is about kit beers. I made a coopers blonde kit and used coopers carbonation drops and it was under carbonated. That batch I left in primary for one week and secondary for two weeks. My second batch was a coopers ipa and I left it for two weeks primary and one week secondary, bottled it tonight with a batch prime. Now my question is as follows....

Does the yeast die as it settles out, or just go dormant?

Because the small amount of sediment that was in the carboy I stirred up into the beer when I primed it hoping that there would be enough mixed with the sugar and alive to make a good carbonation. Will it carbonate properly or just be a waste of time?

I'm also pretty new to brewing but I believe you'll be fine. I left my first couple of beers in the primary for 4wks and 5wks and had no problems. I had very little yeast when bottled.
After 2 wks in the bottle, my beer wasn't fully carbonated. I gave it another week in the bottle, then refridgerated for a week and it was great. It took some patience, but it definitely paid off.

A good proportion of the yeast that settles out is flocculated, dormant yeast. Even after a good few weeks in primary, and also after a cold crash, when it seems like there's a lot of trub settled, there is still going to be yeast in suspension. The yeast cake can be harvested and used for future batches. Check out yeast rinsing/washing and harvesting threads if you're interested.

You should be fine for carbonation with an ambient temperature of about 70*F to help revitalise the yeast.

You'll have plenty of yeast in your beer. You'd have to use a 1 micron filter to get (most) of the yeast out. Yeast are 1-3 micron in size.

You do not want to shake the carboy before bottling. There will be enough yeast in suspension (i.e. did not flocculate) to carbonate your bottles.

All you really did was suspend the yeast cake and sediment. Although it probably did not ruin your beer, it will cause your beer to be hazy. And if you drink live yeast (besides weizen) you will not like it.

In other words, your doing fine. Just dont resuspend your yeast next time. Oh and honestly I leave ales in the primary for ~3 weeks.

Good to know. There was very little sediment in the secondary and it was only slightly hazy when I stirred it. I am not too worried about the yeast. It will settle out in the bottle again before I drink it. I have it conditioning in my room which would be an ambient temp of 22-24 degrees (70-75 F).

I was mostly worried because my first batch didn't carbonate very well and it was really clear at bottling. I didn't like the carbonation drops they seemed too inconsistent.