What's at issue here? There are legal, personal, and ethical reasons for doing a number of things here, and as IT personnel, it's your job to plot a path through this mess. You know there are wrongs, and those should stop. You know what would be right, and it's your job to ensure that happens (and sooner, rather than later).
So what do you do? Well, first off, keep everything on record, via e-mail BCC'd to one (or twenty, if you're paranoid) personal e-mail accounts you set up specifically for the purpose of documenting this.
First, you need to inventory what's going on. How many machines do you have, what software are they running? How many licenses do you have? Make a database of all of your licenses.
Secondly, it's your obligation via your position (as well as ethically) to let your employer know what's going on. It needs to be completely transparent to them what the current situation is, and how illegally the company is currently operating. If your boss is an unethical jerk; it's your job to tell HIS boss (if it's that large of a company) what's going on. Tell everyone who might get their ass kicked by the BSA what's going on. Yeah, they probably should know. But it's your job to make damn sure they do.
Third, propose options for fixing what's wrong. It'd be a lot easier on you to report them to the BSA, collect a fat check (maybe?), and go to Tahiti. It'd take some real gusto to actually break out a bucket o' elbow grease and fix this stuff. Find out what software you have and do an analysis if you really need it all. You mention Office, Acrobat, WinZip, and AVG, most of which have previously mentioned FOSS replacements. No, they won't be as *good*, but they will be legal, and in most cases they will do all that you need them to do. They will also be less likely to contain malware / keyloggers / other crap that I'm sure your employers would rather forego dealing with. You might even be able to find some commercial, non main-brand replacements for some apps (I recall some companies making PDF creators / editors that don't cost an arm and a leg, and will probably get the job done).
What were you hired for? Your company counts on you to:
1) make sure everything works so that business can continue
2) make sure everything is legal & licensed
3) minimize the cost of running the IT department while maximizing value
It's that last step that can lead people to consider employing unethical practices. We all want to save a couple of bucks. If you want to do your job and be able to look yourself in the mirror, you need to do everything you can to enact a swift transition to new practices. By all means, cover your ass by documenting your efforts and if need be quit your job & report the company, but try to do your job as much as you can before you resort to that drastic measure.
In my opinion, you'll be a hero if you can suggest some sort of a compromise that won't kill your company, but still make them legal. No, you can't go and uninstall every illegal piece of software right away, because the company will go under from a loss of productivity. But do try everything you can to fix the situation, write up a nice and pretty report that you can cc to all company management that gives them a good idea of what's going on (make sure to give them a summary that explicitly states the urgency of the situation), and for jc's sake put all the work you do on your resume.
In the end, you're doing this job for yourself. Don't accept the morals of the lowest common denominator, try to raise the lowest common denominator. Do some real good and bring another company up to compliance. People like you who want to do what's right are rare, and people who are willing to work 16 hour days for a week to get this done ASAP are even rarer. Work your butt off for a little while to bring more than just problems to your management, help them by bringing them solutions too.
What'll make you feel better? Sinking your company and dozens of jobs because of ignorant management, or convincing that management that you're doing everything you can to save them money and act legally, potentially saving the company and all those jobs from disappearing with one BSA report?