Assuming home schooling would be the only education for my child - how much time per day should I commit teaching him? And how much time should kid work on his own ("homework")
It depends on the curriculum you choose or the curriculum you make up (obviously using a pre-made curriculum will cost you less time.) It depends on whether your child needs more or less help with a lesson or subject. It depends on how well your child works independently. It depends on what you want your child to learn (it takes more time to teach your child to think critically than to regurgitate facts.) It depends on your style of teaching (are you hands on or hands off?) It depends on how many subjects you want to teach (my kids started learning Latin in the third grade. Obviously this took more time than friends who were not teaching languages at this age.) It depends on the grade (the early elementary grades take very little time; the older the child gets, the more time it takes. By high school, it was more than a full-time job.) It depends on how many kids you're teaching at the same time. It depends if you intend to teach year-round or only for 9 months (most homeschoolers I know/knew taught year round with more frequent breaks throughout the year or a half-schedule in summer.)
It depends on so many factors that it is for all intents and purposes primarily opinion based.
For my kids, homeschool started at 8:30 and finished at noon for grades K-2, with the most time given to my reading stories and teaching them to read. From third grade on, add 30-45 minutes/day for each year, more if you're teaching more than the basics.
Not all of that time is spent teaching/sitting with the student. But you have to be available. In the advanced grades, you will need to teach something you might not know yourself. In this case, you need to learn the subject yourself first, or use tutors, co-ops, etc. It all takes time.
Edited to add: My kids didn't have "homework" in the lower grades. They did have occasional projects they had to organize and work on independently. In the more advanced elementary grades, they got homework in some subjects (not the majority). In high school, they had homework in most subjects.
There are many different styles of homeschooling, and many different learning styles of children. Highly independent children who follow a radical unschooling approach may require hardly any direct input from their parents. I've also known very intense homeschoolers who spend crazy amounts of time on direct instruction with children they felt were behind, upwards of 8 hours daily. You can take as much time or as little time as your family needs to meet your educational objectives.
The homeschooling families I know (mostly elementary age) tend to do around 2 hours of direct parent interaction and around 2 hours of independent work, depending on the child. On weekends or holidays when I'm home to help with a special project or field trip, it can go a lot longer, but that's because we're having fun.