Yes, you can use "nicht mehr" in that way. But people might think that you are trying to imply that you went to the cinema several times before and now you cannot go any more because of your test. So if you just want to say "I was planning to, but now cannot go" without making any implications about the past, then a better way to phrase your sentence would be: "Ich hatte einen Plan, ins Kino zu gehen, aber ich kann doch nicht gehen, weil ich eine Prüfung habe". But like I said, your sentence is totally correct too and people do use "nicht mehr" for that.

Yes, that's fine. (I'd say "ich hatte vor" instead of "ich hatte einen Plan", though.)

In terms of grammar, your attempt is perfectly fine. But a native speaker would never say that. A very common alternative to what my fellow commentators wrote would be: Eigentlich hatte ich vor, ins Kino zu gehen, ... Eigentlich wollte ich ins Kino gehen, aber ich kann doch nicht mehr, weil ich (morgen) eine Prüfung habe. We often start sentences with "eigentlich"/"actually" when we want to use "doch nicht mehr". You could say "aber ich kann doch nicht mehr gehen", but it does not sound good to use "gehen" two times. In most cases, we would add a date to emphasise the urgency. You probably have plenty of "Prüfungen" and I doubt that you mean the one next year.

Another option is: "Ich hatte geplant ins Kino zu gehen,..." The "nicht mehr" works out fine.