I am doing my senior year from a different high school.

So I was wondering if Common App would allow my LORs to be from teachers from my past high school. Also, the counselor recommendation. Thank You.

Yes

The counselor rec would be from the current HS. S/he may or may not opt to include comments from your old HS.

Great!

@skieurope I also had another query. I have planned to start working on my common app July 5 onwards. Is it ok? I am really not sure when is the optimum time to start. Thanks.

You can also ask your old HS counselor for a LOR.
What you need to do straight away is to ask your new school how they are going to convert your old school’s grades. My daughter moved to a new school when she was a junior. It took a lot of negotiation and meetings to get her old grades converted. If you are not proactive about it, you may have lower GPA therefore lower class ranking.
You should start on your essays as soon as possible. Make sure you are using the current application for the year you are applying or your input may all get deleted when they upload the most current application.

It’s fine. Note that the info transfers over to the 2017-2017 version on or about 8/1, and most college essay topics are not avaliable until then anyway.