Is it recommended to just stay in gear with the clutch in while at a light, instead of downshifting to neutral?
For safety reasons it is recommended. It's much faster to get your bike moving and out of the way if an emergency situation were to occur.
I always shift into neutral at a light because my clutch is much more firm than a 250 or 650. It can wear your hand out just holding it in for long periods of time. But I just put my bike in neutral at stoplights for comfort. I have had both bikes I've owned do what you've described. It's pretty normal behavior although it usually doesn't happen often. I just pop my foot two or three times on the shifter at a rapid pace and sometimes that works or just move your bike forward an inch or two and try to shift. The second method works every time but the first method is the quickest way but doesn't always work. Yet again all can be avoided by just leaving it in gear to begin with. It's all personal preference.
Mine has to be in neutral to start
Nah, it's not that bad
Even if you have the clutch pulled in and brake applied? Sometimes bikes have safety features like these. Also most bikes won't start in gear if the kickstand is down.
Even with the clutch in it won't, now thinking about it idk if brake was applied or not. And no, it won't with the kickstand down at all.
