I would like to know if is possible to change the front
page search page on this theme,to add field for choosing the category
also and in the field Location just to be listed Cities?

Also to show the "What are you looking for?",city location and category fields on
mobile phones?

Is it possible to remove fields like City area,Address,Zip,Transaction,Condition from
the "Publish a new listing?" page?

On the front page is it possible in Browse categories to display only
subcategories of certain category,because I plan to have classifieds only for
one category,if you understand me?

Is it possible to remove paragraph "Most favorited listings by users"?

When viewing ad,is it possible to remove tab "location" and message seller field?

Is it possible to add new link on the main menu?

Can I get back office demo of this theme to see how it looks like?

Removing some block is usually very simple, comparing to create new fields or areas.
For most of your questions, answer is that it is possible of course, but you will not find such setting, you will have to do modifications.

Btw. have 1 category does not make sense, create subcategories as parents.

Do you offer those modifications?

And can I get the back admin demo to see how it looks like from admin point of view?

Yes, you can get quote at [email protected]