Hopefully, with the release of Episode II and III we will have some decent answers to these questions.

A.  How come Obi-Wan doesn't "seem to remember ever owning a droid" in ANH?

Simple. He never did. Why he failed to recognize R2-D2, since he saved the ship he was on in TPM is beyond me.

There are only three possible explanations for this: 1) Artoo's design was so common, he wouldn't think that droid was really Artoo. 2) He didn't want Luke to know about Artoo's history, which would bring up a lot of questions. 3) His memory is bad despite his Jedi training (yeah, right.)

The true answer: 4) George Lucas probably didn't care enough to fill in that bobo.

(Note: At no point in TPM does Obi-Wan ever see C-3PO.)

B.  Who trained Obi-Wan? Qui-Gon or Yoda?

In ESB, Obi-Wan's 'spirit' tells Luke to seek out Yoda, the Jedi Master that trained him. It's clear in TPM that Qui-Gon Jinn is the Jedi Master that trained him. Once again, Obi-Wan has caused some confusion.

There are only two possible answers to this question: 1) He didn't want to tell Luke the whole story (including that of his father) 2) Yoda DID train Obi-Wan before Qui-Gon did. 

The most probable answer: 3) George Lucas hadn't created Qui-Gon Jinn at the time.

C.  How come Darh Vader doesn't recognize Artoo/Threepio?

At first I thought Lucas had thought this one out, but after Episode II, nothing makes sense. Anakin, as a child, built C3PO himself. Artoo helped Anakin finish the pod.

You could chalk forgetting after all those years, but then 10 years later, he is reintroduced to Artoo and Threepio (along with Jar Jar and Padme)

Considering that Anakin created C3PO, it seems strange that he doesn't recognize him in Cloud City in ESB. Of course, Threepio does look VERY different in both movies, doesn't he? In ESB, he's in pieces hanging from Chewie's back.

Keep in mind that there seem to be plenty of droids that look like 3PO. In the original trilogy, there are a few droids that look like C-3PO, notice that they are Imperial droids. Perhaps Vader felt a bit homesick? :)

<<Special thanks to jk286.>>

D.  Fine. How come Threepio doesn't recognize Vader OR the fact that 'Master Luke' and his creator share the same last name?

This one's more complicated. Once again, Threepio was left incomplete in more than just his appearance in TPM. His programming had yet to be completed. If this plays any part in his not remembering his creator's name (ie. memory rewrite,) it will be explained in either Episode II or III.

As for Threepio not recognizing Vader, besides the obvious fact that he wears a mask, it's clear that Vader's identify was not known to many (it even looks as though Uncle Owen and Aunt Beru only know that Luke's father was some kind of hero/adventurer... Beru didn't seem to mind very much that Luke had "too much of his father in him" in ANH.)

E.  Considering what a popular kid Anakin becomes in Episode I, how come nobody considered the fact that he might be related to Luke?

Not only does Anakin Skywalker, the slave boy from Tatooine become a famous hero, but Luke Skywalker, farmboy from Tatooine ALSO becomes a hero in similar fashion. Why didn't people put two and two together? It's obvious Darth Vader did between ANH and ESB, when he discovered Luke. 

One possible explanation is that 'Skywalker' is a popular name in Tatooine.

(Still, how many Skywalkers have single-handedly destroyed large space stations, defeating the overwhelmingly large odds? Come ON, George! Give us something to work with here...)