I would say, go ahead and start strength training now, don't bother waiting unless you really need the adjustment time for you mentally. The physical approximately three week adaptation phase will pass and there's no real reason to wait to be adapted because you will likely need to adapt again when you start working out anyway. Working out on keto is similar to fasted training due to the low insulin levels. You can adapt to both at the same time just as easily.

Yes, you can totally lose fat and gain muscle on keto. I wouldn't say it's easy, but I wouldn't say that's something easy under any eating plan, but it can totally be done.

For me personally I tend to do well with carb cycling (higher carbs on strength days) and eating those carbs post-workout. But everyone is different. Some people find better performance getting their carbs 45 min before strength training. Some people are able to build muscle even eating zero carb. So, try something that fits easily in your lifestyle, see if it works, and don't be afraid to change things up and experiment a bit if you don't get the results you want.

In a healthy metabolism glucose and ketones are both burned, pretty much all the time in varying ratios...sometimes more glucose, sometimes more ketones, just based on what and when you have eaten. When you are active your body can burn through a great deal of glucose in a fairly short time depending on the intensity of the exercise. Strength training will also trigger additional storage of glucose (and water) within the muscle fiber, so again, lots of glucose can be shuttled away pretty quickly given a high level of muscle stimulation. Ketosis is high levels of ketones in the blood, which doesn't happen instantly, and also doesn't vanish instantly.

I use keto to control my migraines, I have had no recurrence of symptoms even with carbs as high as 100g on strength training days.

Specifically, I add between 35 and 50 g of carbs on strength days...I tend to add less for short routines (like 30 min of weights) and more for longer routines (like 60-90 min of weights), and I do adjust my workouts every month and those adjustments drive changes to my eating (both calories and carbs) so for me, it's kind of a constantly adapting process in my universe.

Because when it comes to exercise I have the attention span of a toddler...and I won't do it if it's boring, so I keep changing things to keep it interesting.