1. By saying "ongoing", you mean something that is still true or going to happen?
Yes/maybe. The details are very complicated, and depend on the lexical aspect of the verb, among other things.
It's better to think of this in terms of usage choices than grammar rules: when speakers choose to use words in a certain way, they are presenting the information in the sentence according to that grammatical model. If someone says "I forgot [PRESENT]" then they are indicating that the "[PRESENT]" part of the sentence is still ongoing or relevant in some way. (That's why you can "bend" the rules in context, by the way.)
2. If something is "ongoing", does it make much of a (stylistic?) diffeence if we change the verb form (e.g. from "have" to "had" - as in the already mentioned example) or not, saying everything in the past tense?
Very little difference. Aside from substitutability (that is, if you were talking about a different situation, whether you could use either of both of those same forms then), the two forms are basically equivalent in the contexts where they both could be used. But two forms are
never equivalent, because they are associated for speakers with other situations in which they might be used, or even just the way other speakers have used them (in specific cases) in the past. Here there might be a very slight sense of formality with the past tense (just like other cases where using a marked/past form is a bit more formal), but there's no substantial different meaning to my ears. It does, however, give a suggestion that the speaker is or is not thinking of present-relevance.
In short, instead of asking what the rules are, try to understand
why speakers would make the choices that they do. In this case, it's simple: they must choose one form (aside from some kind of awkward circumlocution like "had and still have"!), and therefore they choose either to keep it in the past tense (the "simplest" in a way), but removed from present-relevance, or to highlight the present-relevance by using the past tense. I suppose it might be implied that using the past tense is
instead of indicating that the event is ongoing, so you'd want to use the present
if possible, to avoid that implicature, but I don't get a strong intuition like that with these sentences.
To make things more confusing, you could also say:
"I forgot (that) I was going to have..."
(But that might suggest it did not or will not occur. Future-of-the-past has its own complications.)
Or instead of the present (with essentially the same meaning) you could also say:
"I forgot (that) I will have..."
Honestly, I'd guess that these forms might even be mis-used by speakers (careless usage that doesn't literally line up with what they meant)
just about as often as they are used with any specific intention in mind to highlight relative timing of events. English is fairly loose about permitting this sort of thing (here, being able to mix past, present or future forms all in the same sentence!). The decision might simply be whatever seems most natural, i.e., something like what they heard most recently in their input. But sometimes there is a slight distinction, e.g., to highlight present-relevance.
I don't have a more explicit answer for you about this.