Will coaching for the IIT-JEE in Class 11 affect the Class 12 boards?

Only you can assess it.  Essentially the syllabus is same. The concepts would be explored in a more elaborate manner at IIT JEE coaching. Most students are not able to manage time to cover it all. Lot of time is wasted in traveling, settling down for classes, repeating same topics and worrying about future. If you are still struggling with basics, consider getting one to one help with a dedicated tutor. Once you cover basics, you can go far coaching classes.

It completely depends upon you . How much you can tackle to prepare IIT jee along with boards . I know many people who performed well in both board and IIT jee . There are also examples of people who have mediocre results in board but excellent performance in jee -advanced . If you manage to get times for both , then it will not be problem.

No, ideally it shouldn't. A lot of students do join coaching for JEE right in 11th std only. Initially, when you enter a coaching institute & try to manage it with the everyday activities, you might not fit in well & feel weird of what's happening, but eventually, you figure out ways on how to manage your coaching and daily activities along with school. Your coaching is meant to help you precise JEE examination. The topics and chapters are taught excellently at the coaching institute because you will appear for a national level examination competing with some of the top brains of the country.
Your board exam syllabus by default gets covered in coaching institutes, but the order might differ. It is not necessary for what you are taught in school might be the same as you're prepared at coaching. The teachers are different, their teaching styles are different & your learning pace is different. So, don't worry! If you know how to manage time and are regular with self-study, it won't bother much. But if you aren't able to cope up with coaching & school simultaneously even after 11th, I would suggest you enrol in online coaching rather than offline methods & do self-study.