What exactly is an American politician presented as “Congressman John Doe?” Since Congress consists of the Senate and the House of Representatives, I guess he could be either a Senator or a Representative; yet it is my impression that Senators are more common to be referred to as such, so it could be that a “Congressman” is more likely to be a member of the House. IS there any common practice on this?
Your impression is accurate. “Congressman XYZ” is almost always a member of the House. Senators are called “Senator XYZ” and I don’t think I have ever heard a Senator called “Congressman.” But it would be technically correct, as you point out.
It’s a member of the House of Representatives. I am not sure how this form of address originated but its as good as “Representative X”.
Yes Senators and Representatives are all “members of the Congress” and therefore “Congressmen” or “Congresswomen”. You just wouldn’t call a Senator “Congressman” unless you want to be rebuked by some junior aide or the Senator. With a couple of them, you might get a punch in the nose.
Similarly, in the U.K. and in Canada, a member of the House of Commons is always referred to as a Member of Parliament, or M.P. Technically, members of the House of Lords (in the U.K.) or of the Senate (in Canada) are also members of Parliament, but the term M.P. is used pretty much exclusively for members of the Commons.
Yes, usage is inconsistent. “Congress” means both houses of the federal legislature, but “Congressman” means a member of the House of Representatives - unless the context is both houses, and then it includes the members of both collectively. When referring to a specific person, the title “Representative” for a House member is more common. If an abbreviation for a title is needed, it’s always “Rep.” or “Sen.”.
At least there’s no room for confusion, even of gender, with “Senator”.
Serious Part
A Houseman is the male version of the term for a female household cleaning servant, Housemaid.
So, that term was never used.
Other
BTW–I prefer the gender-neutral term, “Congress-critter”. 
Well, yes and no. Technically, -or is a masculine ending. The equivalent feminine form would be Senatrix, perhaps?
Senatrix Clinton. I like that. 
now, why does that terminology make me think of Hilary in leather…
Ain’t so in English, you heathen furriner.
I believe there was discussion when Margaret Chase Smith was elected senator from Maine, as to whether she was to be titled “senator”, or something with a feminine ending (and no, she wasn’t the first woman to serve in the US senate, but she was probably the first one to achieve a high enough profile for the question to come up in the national press). The consensus reached was that “senator” applied to both genders.
“Congressman” has always created clumsiness wrt gender, and is part of the reason “representative” has become more prevalent. When speaking informally, “congresscritter” is more satisfying to use, but, for some reason, those particular denizens of the Washington zoo seem to bridle at being so addressed …
Not in English, but in French we have madame la sénatrice L’Hervieux-Payette.
I recall stories about the appointment of the first female cabinet member in Spain requiring their version of the Academy to declare that “La Ministra” was acceptable.
Well, not for most of the year. But on Stockwell Day…
Um, try executor/executrix?
Or we can ask Merriam-Webster Online. 
Note the part about “Middle English.”
I’ve thought of a better one that Senatrix Clinton: Senatrix Boxer. REALLY fits.
Note the part about “from Latin.” 
Pedants, snobs, and general troublemakers keep trying to turn English into Latin. It doesn’t stick.
Thath’s the only example still extant that I know of, “aviatrix” having been left behind us. Even “actor” can be gender-nonspecific now.
France and Quebec have taken opposite approachs to this problem. In France women are simple refered to with the masculine form (madam le president) while Quebecers either use female forms that were once used to apply to the wife of a man so titled. So women in Quebec use the eqivalent of terms like; doctress, ambassadress, presidentess, etc.
“Congressman” also gets around the confusion that might arise as to whether the critter in question was a member of the US Congress or some state legislature. Several states have a “House of Representatives.”
Um, most of the current English language is the result of Latin or French roots. Very little of it is good old Anglo-Saxon. If you want to eliminate all remnants of what we still use from Latin forms, I suppose you don’t like octopi?
Like it or not, -tor still has male connotations, there are words we use (I’ve pointed out two of them) that still take the -trix ending for females, and that’s part of English. Maybe not for much longer, but still there.
<going down with the swords swinging!!!>
On Edit: I thought I had given two of them. Guess I forgot to throw out aviatrix, which, while not in common useage, is still a valid term. We just don’t call men aviators anymore. (Pilot is sooooooo boring by comparison!)
Dominatrix is still in frequent use in, er, certain circles.