It seems like a fun idea - come up with novel ways of creating advertising, and potentially get a prize and some publicity for your effort. In exchange, Firefox gets an advert and some publicity for itself.
This might ramble a bit, so bear with me:
Well think about it, we already have an enormous Google advert in every copy...
You mean the search box? That can either be completely removed (right click, customize, drag and drop) or set to use another search provider (search dropdown, take your pick, or add more from AMO)? That has a simple UI to remove Google, the same as any other search engine?
2) The Mozilla Foundation is heavily funded by Google. I use Google all the time, but I am very wary of it... its just another company, we could use an open source search engine that Firefox could be linked to rather than Google. Google is more interested in advertising really than anything else. Mozilla buys into that.
The MoCo gets a lot of money from the Google search deal, which percolates into MoFo to pay developers, buy/run architecture etc (the Radiant Core theme for Fx2 wasn't free and the money doesn't come from thin air), but it's not like Mozilla is turning a massive profit from all of this, or is some kind of Google subsidiary like you imply. Don't get me wrong, MoCo gets millions of dollars from Google every year, but they're not selling out in the way you reckon.
If there was a decent OS SE then there'd be a decent argument for its inclusion, but AFAIK there isn't one that can come close to Google (and if there is, why doesn't it already have a search addon for Fx?). Google aren't the only people that contribute to MoFo either - IBM and Sun have an input as well, and I
think the Malware protection that's in Fx3 isn't from Google.
Mozilla doesn't try to shut out other providers either (a la Live Search in IE7, though you can add others they aren't there at the start) - Wikipedia and Yahoo are there by default as well...
If you want to be paranoid though, you'd be better served looking at the phishing protection, which could send your browsing habits to Google - though the active checking (i.e. check with Google for every site, rather than with a cached file) is switched off by default, before anyone gets Tinfoil Hat Syndrome.
4) They are (or already have) dumped/dumping Thunderbird into its own project, unfunded. So it will die. As another one of its classic XULRunner type applications, why would it do that?
Actually, it's getting a bunch of cash (that it could hardly say it earned) to start up MailCo, and until they left MoFo (of their own will AIUI; they have said they intend to continue doing what they were as volunteers rather than as paid staff) there were two paid devs who spent the majority of their time on Thunderbird.
Sunbird/calendar/Lightning don't get any cash from MoFo either and certainly haven't died.
I'd be wary of what you read about MoFo/Co/the split with Thunderbird to form MailCo because a hell of a lot of it has no basis in fact and is largely trolling.