how long can I stay in an open primary fermentation?

You'll have no issues with an open fermentation in terms of contamination from the open air if there is still a layer of CO2 at the interface of the liquid and the air, which will be the case during active fermentation. I'm assuming that your container is covered with something, either plastic or a lid and ferm. lock. If you are completely open to the air (why?) then chances are very high that you will have a disaster due to air-borne bacteria regardless of fermentation activity.

There is probably a grace period of a few days after the end of fermentation when you can bottle directly without much danger of bacteria getting into the beer (assuming, again, that you have a cover of some kind over your container). Keep your bucket away from forced-air heating vents and/or drafty areas in the house.

But the other issue will be avoiding getting excessive haze and sediment into your bottles (ie: trub, including yeast and hop bits), unless you use a fining agent to accelerate the clearing process, which can otherwise take over a week.

One way to avoid getting some of the worst gunk into your bottles is to put a mesh filter on the end of your siphoning tube, the end that goes into the bucket. This will prevent the largest bits of sediment from getting into your bottles, but you will still have a lot of haze particles in your bottles if the primary has not been given time to settle out; especially in the last 6 or 7 bottles as you try to siphon off the last of your green beer from the bottom of your primary.

Another option to extend the air-free time while fermentation subsides and the clearing process takes place, is to use a CO2 injector to apply a layer of CO2 to the surface of the beer once in a while, until the primariy is relatively clear (but, again, usually not before 1 week after fermentation has ended). You can get handheld injectors that take 12g or 16g CO2 cylinders. Just make sure the CO2 in the cylinders is food-grade ("100% CO2 only"), and is NOT meant for BB-guns; those will contain oil.

Your mileage may vary...

If it was my brew, I certainly would not give up on it (yet). I would gently set it out and let it settle for a few hours, then rack to another container, being careful not to pull anything (like glass) into the second fermenter. (Leave more than normal and make sure and use the diverter on your siphon.) Then you can triage the carboy and clean up the mess. However, the reason I suggest this is to avoid losing all your beer if the carboy continues to break.

I would not be too concerned about an open ferment. I have fermented several beers without a lid. I tried it a few times out of curiosity more than anything, but I kind of quit the practice a year or so ago. However, I didn't quit because I thought the beer was bad, just because I didn't think I was gaining much for the risk. Having said that, I've had batches with off flavors with "normal" practices and none of my open fermentations ever gave me any grief. They typically fermented much faster and produced a more significant krausen, but also produced more esters, but overall, nothing unexpected.

In an IPA, I doubt you would notice the difference, because of the dominant hops. The hops will also help stave off bacteria. Good luck!

There's basically no need to do a "secondary" fermentation. Time in secondary is just as good as time in primary.

You can go straight from primary to bottling if you like, so long as fermentation has actually finished. Once the krausen falls further, and you get the same gravity readings over the course of 2-3 days, you can bottle straight away.

However, if your glass carboy really did break, I would be very worried about glass shards in the batch, which are going to be nearly impossible to detect and remove. I'd dump the batch, just to be safe, and obviously toss the carboy. As well, I'm not sure how you're going to safely lift the broken carboy in order to siphon into the bottling bucket.

As well, I would cover the top of the fermentor with aluminum foil to prevent anything from falling into the beer.

Good luck.