With all due respect, I think you are being a little too laid back about this issue. When you say that inflation will slowly work off a bit of the debt, you do realize why right? It is because inflation is the decrease in value of the dollar, meaning that the USD's buying power decreases. This means that with inflation the government is essentially taxing the people and anyone else who holds USD.
As for your professors, please remember that economics is all based on theories and your professors' ideas aren't necessarily truth. You should really read other economic theories before blindly following one (the one you learned about is most likely Keynesian from your descriptions.)
The bailout is not the best option we have because it goes against everything we stand for. Failing businesses is a part of capitalism and by not letting anyone fail you are hurting the capitalist system and in fact, pushing the socialist one. Also, you brought up the point about how the government will get the money back in the companies it invested in if they don't fail. But what if they do fail? Who will lose their money? We, the people will.
Well of course I don't believe that inflation should be driven up to pay off the debt. That would just be crazy. They might as well increase taxes and pay it off the old fashioned way.
As it stands, we will always have a little bit of inflation. Believing that inflation won't exist is truly living in dreamland. There's about 2-3% of inflation that keeps us healthy and ultimately encourages people to keep their money flowing through the system. Deflation has also proven to be truly devastating, because it quickly snowballs out of control and kills the motivation for investment. Why invest the dollar you have if it will be worth $1.10 in a month?
As far as my professors, I'm not exactly following anyone in particular. It's a basis that's been built from three teachers in different areas and private research and discussion in other forums. I'm aware of Keynesian. That model basically guides most of economics. It is wrong when taken to any extreme but right on the whole and as a general guide.
The bailout is the best option we have, because there isn't any other option for a rescue. As you said, the only other option was to let them sink or swim. I agree in principle, but I can't agree practically. It could be argued that the government should have stepped in and stopped their risky behavior. We are forming regulations now to do just that.
It may have been truer to capitalism to have let them fail, but that would have destroyed the economy. We were lucky that it stopped short. A number of businesses would have been utterly ruined if their lines of credit had fallen through. Allowing the banks to fail would have caused untold amounts of damage. It's the same problem with bailing out the auto makers. They are terribly run companies which made bad decisions. Does that mean that we should let 2 million people go out of work by letting them fail?
It was a bad situation and this was probably the best way out. I believe that they bought preferred stocks which puts the government in a good place for recovery in the event that they would still fail. They also have a lot of oversight in the company which makes failure unlikely. I do wish that they would have been a bit tougher on the boards (Britain followed a model I liked a bit better) but that's just a nit pick.
Besides, if the banks still fail, then we're basically doomed anyway.