Everything I've ever read about US IP law tells me that the things which count as "fair use" are very similar to those quoted above for UK law. You'll probably find that there's a wealth of precedent and case law that defines this.
Some games like, "I wanna be the boshi" actually has various copyrighted characters from street fighter, mario, etc. I would look into why they aren't sued yet. ;)
They're freeware games that big corporations don't care about.
Most companies don't just sue because they can. They do it when it makes business sense to do so. Consulting with lawyers takes time and money. Unless they are going to make MORE money as a result of a lawsuit, they're not going to file one.
Freeware games can get away with this because they're small enough to not be a concern.
Which comes back to my #2 option before. Just do this without getting official permission from the copyright holder... and you'll probably be fine as long as you don't sell it. There are literally thousands and thousands of these kinds of games (even more if you count things like ROM hacks and other game mods). Nobody really cares.
Theoretically you could... since you're violating a copyright and distributing material you don't have the rights to.
Realistically, you won't... since it would be expensive for the BBC to file a lawsuit and follow up on something that they probably don't care about.
If you're really worried about legal action... then don't use Dr. Who. Use a knockoff. Call it "Dr. Shoe" who travels through time and space in the "Heartis".
Whovians will get the reference and it'll be 100% legal.
Companies only care about money. If you're not subtracting from their profits, they will ignore you. If you are adding to their profits, they'll try to get you to sign stuff so that you keep it up and don't turn it around.
I realize it is just an example, but Dr. Shoe? I have to admit I died laughing at that one.
i am curious if i do get a cease and desist letter and i take it down will i still get sued?
Usually no, the point of the cease and desist letter is basically a scare tactic to avoid suing because they know how much it would cost. The stipulation of the letter is that if you stop, take it down, and never pick it up again then you won't incur any further legal action.
A cease and desist letter, also known as an "infringement letter" or a "demand letter," is an order or request that a party halt an activity ("cease") and not take it up again later ("desist"), else face legal action.
[EDIT]
Sorry, while I laughed, the fact that you have to ask this many questions tells me you need to drop the idea and listen to Disch at this point. If you are worried that bad about legal action, you need to take the ideas and change it so you aren't using anything that will bring on possible legal action.