I’ve heard people say or write,
“I wanted to inform you that…”
or
“I’m calling because I wanted to invite you to…”
or
“I’m writing because I felt that…”
In context, these are clearly present-tense statements, but past-tense verbs are being used. Somehow it seems more “polite” than present tense, a little more formal, though I can’t figure out why. So what is the reasoning (if any) for this practice, and how did it originate?
It doesn’t seem more polite or formal to me. I don’t think there is a reason or a known origin for speaking incorrectly.
Well, in the second one it’s clear that the caller is speaking about why he decided to make the call so he/she is talking about a past ‘event’. The same with the first, the speaker is telling you why he decided to talk to you.
sorry, ditto the third as well.
I think there is a certain level of politeness to "I wanted to invite you … " rather than “I want to invite you.” Regardless, I think these expressions are just idiomatic and not incorrect (except the third example, which strikes me as awkward). Also, logically, you could say that the speaker “wanted to invite” the person before and IS inviting them now. The desire was in the past, the action is in the present.
Makes sense to me. They felt or wanted something in the past, and are only relaying the message now.
aaslatten, the first one looks awkward as it apparently lacks a formal context, but it’s still not incorrect and could easily be placed into a fromal context, e.g: “I am wrinting because I felt it would be in your interest to know that…”
I’m with aaslatten here–it seems to me that these usages are idiomatic, and that there’s no real reason to be concerned about them.
I looked in Bryan Garner’s Dictionary of Modern American Usage and couldn’t find anything on this particular issue.
Are you sure you’re not mistaking the subjunctive mood for the past tense? The subjunctive is used to express desires, possibilities, things that aren’t facts. “I wanted” and “I felt” are both describing the writer’s desires.
Another example: “If I was a rich man” is not correct – it should be “If I were a rich man”. It’s talking about a desire, not a thing that exists or existed. Frequently, the subjunctive looks like the past, but it’s not.
(I think.)
Your second paragraph is correct, except that it is not really a desire, but rather a counterfactual condition, that is, something that might be but is not.
This aside, however, i really don’t think that the examples given in the OP are in the subjunctive mood. Rather, they demonstrate a sequence of thought on the part of the speaker.
When we say something like “I wanted to inform you that…,” it seems that we are indicating that the desire to inform originated previously, and we are now acting on it. This is, admittedly, little more than a rationalisation on my part, but it seems reasonable enough to me. 
I also think there is something to the OP’s contention that this usage is somehow considered more formal or polite but, like the OP, i have no evidence to back this up.
I think it is probably down to formality, and people who were told not to say “I want” by their parents. Many forms of circumlocution are used in polite discourse. “I wanted to inform you” isn’t ungrammatical; like “I would like to inform you that”, it is merely another polite way of introducing information.
It’s definitely not subjunctive, but I can accept that it’s an idiomatic present tense, not a past tense.
When I hear someone say, “I wanted to inform you…” I feel like answering, “What, you’ve changed your mind, and no longer want to inform me?”
Of course, by then he’s already informed me, so it’s a moot point.
Yes, but when I hear someone say “I want to inform you …” I feel like answering, “Well, go ahead and inform me, if you really want to.”
It’s clearly a description of the process that LED to the decision to act. That process took place in the past, so its description uses the past tense. Probably not the most formally correct usage, but certainly not idiomatic.