I don’t like programs doing things behind my back, thank you very much. If they worked better I might use them, but they paste stupid logos everywhere there is a link in the page.
You don’t say what version of firefox that you have as firefox won’t let any add-ons be installed unless they come from the Mozilla add-ons area or you actually give permission. So I don’t really know why that happened.
When you install avast you can use the custom install function and uncheck options that you don’t require, the Browser protection function is the one that would install the Avast On-line \security plug-in for your installed browser/s.
This Browser Protection function can be uninstalled - control panel > programs & features, avast > Change Remove - from the next window select Modify - uncheck the Browser Protection.
I have Firefox 39.0. Firefox asked me a few times in the past after updating Avast, but not this time.
I don’t believe FF 39.0 did any checking, other than add-ons not being installed from Mozilla (which would be the case for AOS), that too should ask the user.
It is only later versions 42.0 & 43.0 that Mozilla (Firefox) are looking for all add-ons to be digitally signed if they aren’t they won’t be installed. FF 43.0 is even more strict about non-digitally signed add-ons, they are blocked.