While analyzing the effect of level 2 variables on level one outcome variable, only level one related control variables are required to add or also the level 2 control variables are also required to include as controls?
If there are level 2 variables you want to control for then you should include them into your model. When using R (for example lmer function) you have to check wether your level 2 control variables should be included in the random intercept part only or also in the random slope part(s).
Well, I think there is no difference between variables you want or have to control for on level 1 and control variables on level 2. An (maybe very artificial) example: If you are analyzing the effect of the age of a therapist (level 2 variable) on the symptoms of the patient after treatment (level 1 outcome), there are of course level 1- and level 2 variables you could control for. On level 1, you should/could control for the symptoms of the patient before treatment, on level 2 you could control for the years of training of the therapist/ how long he is working as a therapist.
Concerning my second point, using my example: The next question is wether there is a predictor variable on level 1. If not, the years of training can only have an impact on the intercept of the level 1 regression. If there is a predictor variable on level 1, you have to reflect if the years of training have a potential impact only on the intercept of the regression or also on the regression slope.