The URL Bar doesn't size at all/sizes weired
It's very likely that you've got either a "bad" theme (haven't seen one in ages though) or some left-overs in your userChrome.css. Try removing any URL Bar related stuff from that
file in your profile folder. If it doesn't work, the file doesn't exist or you don't know what to do,
contact me and provide the following: Autosizer version, Firefox version, which Theme and in which version plus a link where I can get that theme. I'm sure we can fix it.

Some letters get hidden when I type although the searchbar isn't at its maximum width (applies to input helpers as well).
If you use very "wide" letters or use an input helper tool (especially Asian languages). Autosizer actually auto sizes in advance, making room for another letter even though you didn't type it yet. If your letter is wider than the room Autosizer made before, letters will (partly) get hidden on the other side. This can also be caused by the feed or lock icon in the URL Bar overlaying text.
Increase the offset to fix this. The optimal offset width is the width of the widest letter in px minus 9. For example, for most western languages the capital W is the widest letter, on my system it's 9 pixels in width. Therefore the optimal offset would be 0 pixels.
There are red bars on each side of the URL Bar
- Open the Add-Ons Manager
- Double Click on "URL Bar Autosizer" to open the Assistant (if the option dialog opens, click "Assistant")
- Choose any option and click "Next"
- Click "Cancel"
- Red Bars should be gone
Is this available for the searchbar as well?
Yes it is: Searchbar Autoszier