Synthetic Konjunktiv II
Every German verb has two ways of forming the Konjunktiv II. The first one is what linguists call synthetic, i.e. via inflection. For strong verbs, these forms look like past forms, but with an added ending -e and umlaut if possible:
past indicative - past subjunctive/Konjunktiv II
hatte - hätte
war – wäre
kam – käme
ging – ginge
Note that some verbs have different vowels in the past indicative and past subjunctive: stehen, stand, stünde.
For weak verbs, past indicative and past subjunctive are identical.
Als ich damals in Italien lebte … (=war, past indicative)
Wenn ich jetzt in Italien lebte … (=leben würde, past subjunctive/Konjunktiv II)
Note that although the form is called past subjunctive, it refers to the present and the present only! This will be relevant below.
Analytic Konjunktiv II
Probably because this ambiguity between past indicative and past subjunctive is hard to tolerate, a second way to form the Konjunktiv II has developed.
This is what linguists call analytic, i.e. by combining two words, namely a form of würde plus the bare infinitive.
würde haben, sein, kommen, gehen, leben, …
OP's examples
So far, so good. Let's apply this to your examples.
Wenn er doch heute kommen würde!
Wenn er doch heute gekommen wäre!
The first example is simply the analytic Konjunktiv II of kommen. It is desirable for him to come, but it is not yet known whether he will do so or not.
The second example is the synthetic Konjunktiv II of (gekommen) sein. Above I mentioned that the Konjunktiv II by itself always refers to the present. However, in this case a perfect auxiliary (sein) appears in the Konjunktiv II, which means the second example refers to the past: It would have been desirable for him to come, but he did not.
The other set of examples shows the same kind of difference.
Ich würde gern feiern.
Ich wäre gern feiern.
The first example is the analytic Konjunktiv II of feiern. I have the desire to party, but am not partying at the moment.
The second example is the synthetic Konjunktiv II of (feiern) sein. However, sein plus infinitive is a tricky construction. It could be interpreted as
Ich wäre gern feiern gegangen.
with wäre gegangen as the synthetic Konjunktiv II of (gegangen) sein, with the expected meaning: I would have liked to go partying, but did not in fact do so.
If ellipsis is ruled out, it could be an instance of the construction discussed in this question. The basic meaning is telling someone where you are or will be, or, in the Konjunktiv II, where you would like to be but are not.
Ich wäre ja schon lange feiern, wenn meine Freunde mich pünktlich abgeholt hätten!
=Ich wäre schon lange auf der Party, wenn …
Nothing but a Mix Up?
To be frank, the first time I read
Ich wäre gern feiern.
I thought you had mixed the synthetic Konjunktiv II of sein (wäre) and the analytic Konjunktiv II of feiern (würde feiern). Could this be the case?
Ich wäre auch auf der Party, wenn ich könnte.
Ich würde auch mitfeiern, wenn ich könnte.