See, that is something else and it is your choice to install it or not if you perform a custom installation (which you should do with every software software to avoid nasty surprises).

That is correct

False, it is checked by default to be installed but the user can uncheck it.
Hence why I said to always perform a custom installation.

It only does it with the users permission.
After all, it is the user who has chosen to install it.

Correct, that is kinda how it works.
Completely useless add-on if you ask me.
It only checks prices on sites that partnered up with avast, not the site from e.g. the local bakery around the corner while that can be cheaper than all others.

Some extra info:
The SafeZone Browser is not distributed anymore since the beginning of this month.
avast is gonna release a new browser.
When is still not known.

A good tool for you could be unchecky
I would say have a look at it.