For band instruments, students need to be grown-up enough to handle whatever instrument they're starting.  Typically that's in about 4th grade.
For piano, students need to be able to sit at the piano and focus their attention for at least 20 minutes at a time.  Typically that's at about 5 or 6 years old.  I started at 6 myself.

I love teaching younger kids.  I taught elementary school and middle school for a long time, and I love those ages.  I have a wide range of experience with young kids on many instruments, but as students get older and more advanced, I generally pass them off to a specialist. 

For daily practice, it is necessary to have an instrument at home.  That could be a full sized piano, or an electronic keyboard, at least at first.  

We're all busy people.  Nobody has unlimited time to practice.  Having said that, if you don't make it a habit; that is, if you don't practice a little every day, you can't expect to make progress.  Small amounts of time (I suggest 20 minutes) do more good than long tedious practice sessions.
I have a serious of practice rules that I use with kids at school.  (All but the first one apply to the piano, as well.)
See the video here;
 https://www.youtube.com/playlist?list=PLQRHob9BaKY_E5TJcOMB7SM4i2CQS_yQ_