What if some group of people start sending small transactions from wallet to wallet just to make the blockchain bigger? What if some government start using their computers to move the tiniest amount of Bitcoin possible, from wallet to wallet, just to make the blockchain REALLY big? Is this possible?

And also, is it possible to generate large amounts of transactions so that one winning node can't include them all in one 10 minute mining process?
The current maximum size of a block is 1 MB.  Current block sizes are about half that, so the absolute worse case scenario is that the block chain grows in size twice as fast as it does now.  That's not particularly scary.
It's easy to create more than 1 MB of transactions every 10 minutes.  If anyone does that, some transactions can't be included in blocks.  However, at present, there are almost always more transactions available than miners are willing to include in blocks, so we're already at the point where miners need to choose what transactions get included.  

For the most part, miners choose based on fee per kilobyte of transaction data, so transactions that pay more fee/kB get included in blocks sooner.  If someone wanted to, they could create large transactions that paid large fees in order to raise the transaction fee the rest of us would need to pay.  Since that would cost the attacker money every block without actually preventing us from using the system (if we were willing to pay the higher fees), it's also not particularly scary.