I'm a long time user of Avant, as well. Just started trying out Firefox yesterday. One of the problems with Firefox is that it comes with almost no options in GUI off the get go, which is in stark contrast with Avant which has a multitude of functions. I haven't found any extensions that allow for any of the features you're looking for, unfortunately.
However, here's a list of some of the extensions I've found for Firefox that make it closer to Avant (and in some ways, surpass it):
Tabbrowser Extensions
Given that Firefox's tabs are pretty much useless after installation, this extension is a must. It gives you all the functions you're used to from Avant. Single window mode keeps all the web pages as separate tabs in one window. And you can set the tabs to close when you double-click them (a la Avant). You can also drag and drop your tabs to reorganize them and customize other tab functions more than you could with Avant.
All-in-one Gestures
This little beaut restores the rocker function in Avant (hold right click and press left click to go back a page and vice versa) and also adds in mouse gestures. With these, you can hold the right click and draw to carry out functions. (i.e. you can set it to go back when you draw a lien to the left)
Adblock
Speaks for itself.
Web Developer
Lets you disable images/cookies/aniamted gifs/java/etc. But it doesn't let you disable flash. There's anm extension called Flashblock that blocks all flash animations and replaces them with placeholders. Then you can click on those placeholders to load the flash you actually want to see. However, their site says it's not upgraded for 0.9 yet.
Nuke Anything
This baby is great. It allows you to seemlessly temporarily remove any part of a website. For example, you can remove a huge ass ad out of an article before printing it, or you can hide those ads and pictures that aren't exactly SFW. Everything is restored when you refresh the page.
BugMeNot
This thing lets you bypass those sites that ask for registration by pulling out a user/pass from a user-submitted pool of accounts. Great for sites like nytimes.com
Another function of Firefox I like is the ability to search for text on the page just by typing and then pressing F3 to find the next result. No more hassle of pulling up the find window.
There are downsides to Firefox. I haven't found anything that lets me minimize the thing to the system tray, so it hogs taskbar space. I don't know if there's anything for multiple home pages. There is no full desktop mode with autohide that I know of. You can go fullscreen with F11 but it does not autohide the toolbars, so you have to hide it manually.
I'm still on the fence about which browser I like better. Avant seems to have all the basic functions covered, while Firefox is missing some vital options, but has the ability to be customized by its extensions. It's up to you to decide what browser has the features you just can't live without.