FF writer, here's my response to your question about the following sentences:
"The cat had obviously been lying in the mud."
This is incorrect, as "had" is the past tense of the verb "have", which requires "lay" in this context. The two words -- "had...lying" function as a part of a split verb phrase. The "-ing" suffix merely puts the verb in a progressive or continuous form, so if you are implying that this was happening in the past, then you should write "had been laying", but without context, whether the action is still continuing (which would require present tense) can be unclear to the reader. Appropriate context could be "Sarah looked at the muddy footprints and thought with a sigh, '...'."
If you want to clearly show that the action has stopped, then you would write, "had lain".
"She was just lying around."
Technically, this is incorrect as "was" is the past form of "is", and so "laying" is appropriate. See the above note on the past progressive tense. However, given that the above is probably a sentence one would write in an informal context -- e.g., in a personal e-mail or perhaps in casual speech -- and the fact that "lying around" could be seen as an idiom and not part of a verb phrase (i.e., "is/was lying/laying around") I would suggest that it's OK in informal writing, and that includes dialogue. In formal writing, however, I would avoid it -- just to avoid the chance of offending someone who knows a lot about grammar.
(And to fellow grammar snobs, I will say that although I did just start this sentence with a coordinating conjunction, which makes me cringe in most contexts, it is important to remember that different scenarios have different expectations for adherence to formal grammatical rules. In my opinion, mastering all types of communication is the key to personal and professional success.)

If anyone sympathizes with dear Tom above, answer me this: do you regularly hear things like "was sanking" or "was ranning" or "was thoughting"? No, and do you know why? BECAUSE IT'S WRONG!

(It resides in a realm of things which are so utterly and blatantly wrong that they really require no further exploration... though I can offer same if you so desire.)
As for the
red text: THAT'S NOT HOW USAGE WORKS!! If it's illiterate in one register... IT'S STILL JUST AS ILLITERATE IN A DIFFERENT REGISTER!

I don't think he gets to give an opinion to grammar snobs as if he were one.
