I have often encountered the forms
möchte, möchtest.... Does the verb
möchten exist? Is it some special form of
mögen or an independent verb?
I have often encountered the forms
Technically, möchte is the subjunctive II (Konjunktiv II) of mögen.
However, mögen is special, as it changes in a different way than other verbs do when put into the subjuncitve mood: While with most verbs, the subjunctive II mainly conveys the irrealis (i.e., that whatever is described, is not real), mögen changes its meaning from to like (and some others) to to want¹. Some examples to illustrate this:
Er sagt, dass er meistens Pizza esse. [no subjunctive II]
He says that he usually eats pizza. [No implications are made about the correctness of his statement.]
Er sagt, dass er meistens Pizza äße. [as above with subjunctive II]
He says that he usually eats pizza. [It is stated that his statement was not correct: He does not actually eat Pizza usually.]
Er sagt, er möge Pizza. [no subjunctive II]
He says that he likes pizza. [No implications are made about the correctness of his statement.]
Er sagt, er möchte Pizza. [as above with subjunctive II]
He says that he wants pizza. [No implications are made about the correctness of his statement.]
This is comparable to the difference between the English like and would like.
Such a shift of meaning in the subjunctive II mood happens to some other modal verbs in German (namely sollen, dürfen, and müssen²), but the subjunctive II of these verbs can still be used for the irrealis of those verbs. In contrast, you cannot use möchte and similar anymore for the irrealis of mögen. Instead you have to use a würde construction (which is used for the irrealis of regular/weak verbs and in some other cases):
Er sagt, er würde Pizza mögen. [würde-Form of mögen]
He says that he likes pizza. [It is stated that his statement was not true: He does not actually like Pizza.]
One could therefore say that möchte and similar forms detached themselves from mögen and are to some extents forms of an independent, defective verb, which does not have an infinitve, subjunctive II or perfect form. Analogously, mögen has turned defective and lost its subjunctive II, which has no big consequences, as we can use a construct with würde instead.
¹ To add to the confusion, mögen can still mean to want in some dialects.
² And brauchen to some extent.
There's no verb "möchten", the forms you see are the Konjuntiv II forms of mögen. In fact it's so common that it's often introduced, confusingly, as a modal verb independent from mögen, but that's not correct.
It must be said, however, that the Konjunktinv II is used far more often as a true modal verb than the Indicative. Whereas "ich möchte etw. tun" clearly denotes "I'd like to do sth.", the same usage with the indicative is less common and has a slightly different meaning, eg "Ich mag etw. tun" is more similar to the english "I might do sth.".
"Mögen" however retains its meaning of "to like" in the indicative when used with a direct object: "Ich mag dein neues Auto".
Little children may say "Ich mag ein Eis" to the waiter, till they learn that "ich mag" is not said. The polite form is "Ich möchte ein Eis, bitte". "möchte" is the past subjunctive of "mögen" used as a subjunctive of politeness. But actually it derives from a conditional sentence: Ich möchte ein Eis, wenn es Ihnen recht ist. - Or something like that. (I would like some ice cream, if you don't mind.)
By the way I don't like the German terminology of Konjunkiv I and Konjunktiv II. Here the German terminology steps out of line. In English, French, Italian or Latin there is no Konjunktiv I and II, and learners have to learn what German grammars mean with their special terms.