Oddly, the results are slightly different. Stove top tends to go slightly faster and not have quite as much depth.
No. That high a heat isn't a braise. A braise is cooked "low and slow." Moreover, you don't want "rendering." I presume that by using the term you're trying to say you want "the collagen and connective tissues and fat to melt and flow through the meat." It doesn't actually work like that, and to the extent it does, you want protein denaturing. That process is surest and best at lower temperatures.
You want to braise between 275 and 325. The time differences at either end of this range are large, while the quality difference (nod to the lower temp) is fairly small. 300 is an acceptable compromise.
Searing them, the same way you learned to sear steak is The Way.
Heat the pot. Add oil. Allow the oil to come to temp. Add the seasoned meat. The meat may or not be dredged or dusted in flour as part of its seasoning process. Allow it to stay where it lands without turning or moving or otherwise being a buzz-kill until it's completely browned and ready to turn. Turn and repeat until all sides are browned. Everyone else is heterodox. Death to the heretic! Now, Go Ye Forth and Spread The Way.
Seared brown is what you want. More is not better, it is worse. Less is not better, it is unacceptable. When the ribs are brown, remove them. Pour off any more fat than necessary to brown the aromatics (mirepoix). Brown them quickly. After they've taken a little color -- in other words, browned and not just sweated -- push them to one side, and add a little tomato paste to the empty space on the pan. Let that brown a bit, then stir the paste and vegetables together, then cook the vegetables for a couple more minutes until the paste darkens. Then add a little liquid to deglaze, return the meat to the pan and add the remainder of the liquid.
We've hit the "it depends" barrier, again. Bet you didn't see that coming. There are essentially three ways. First, pass the liquid through a Chinese cap and squeeze the aromatics. Second, use a thickener at the end. Third, use a thickener at the beginning.
Third: Before you brown them, season the ribs, then dredge or preferably dust them in flour. Shake off all the excess. Brown the floured ribs -- most of the flour will come off during the browning -- which is what you want. Meanwhile the meat's juices will penetrate the flour and brown properly.
Shortly before the braise is done, make a medium brown roux (about the same color as peanut butter) with oil and flour or butter and flour. When the braise is otherwise done, you can remove the meat and vegetables, then thicken the liquid with the roux. Thickening with a roux is always done at the boil.
Or, you can make a slurry with cornstarch and water, and use that to thicken. Cornstarch gives you a little more control, but a roux, because it's cooked, will taste better. It's also the most robust and will serve you well as your cooking advances.
Another alternative is arrowroot. Arrowroot works well with very acid (lots of wine) sauces, and thickens almost instantly -- which makes it easy to judge the proper amount. However, it loses its thickening power pretty quickly and is also a bit on the delicate side. Not good if you're planning on leftovers.
At any rate, the key is taking out the meat and vegetables and sieving the sauce before service. This is one of those techniques and "attention to detail" things that separates the excellent from the good.
First: When the braise is almost but not quite done, remove the meat with tongs and set the ribs aside. Empty the cooking pot by passing the liquid through a coarse sieve or Chinese cap. Depending on whether you're using a sieve or a spoon, use the back of a spoon or a "pusher" to force the juices and "structure" out of the veg and into the liquid. This results in a light gravy, at best. However, you have a chance to evaluate how much more thickening you need before adding any additional thickeners.
Which brings us to another key: Discarding the cooking vegetables -- which have become flavorless, discolored and ugly. If you want vegetables as part of almost braised or long-simmered dish, replace them with fresh and, if appropriate different. Cut the service veg neatly, and size them so they will all finish at the same time along with the meat. Better, cut them in attractive shapes and sizes and blanch them off so you can put them in as part of the plating process.
Stock if possible, but it won't make a huge difference. Watch out for salt.
First rule: Whatever works. But a standard, classic, real, French, Escoffier, Julia Child's, etc., mirepoix is 2:1:1. Don't let that influence you though. I'm just sayin' know the rules before you break them. Use what tastes best.
Use tomato paste at the beginning for some structure. Using canned tomatoes and cooking them as part of the braise, then pressing them through a sieve will give you lot of thickening. See above. If you use a flour dust or dredge on the meat, you may not need any additional thickening.
Scrape as much fat off the back as possible. It's not suet, but it's actually pretty good beef fat -- useful for all beef fat purposes.
Probably didn't cook long enough. Keep a plate by the stove, take a rib out a little before you think it should be ready, put it on the plate, and use your fingers to see if it's tender. Or, you could wimp out and use a fork. If it's not tender, it's not done. If it's just tender, it's done. Cook it too long and they get stringy. Not good. Once it's done, get it out of the oven or off the heat (or whatever), and it will hold indefinitely. In fact, it gets better the longer it coasts. This means allow extra time to cook and if you don't need all of it .... fine.
Hope this helps,