I don't know if you mean a feeling of cold, or if you mean a sickness. In English, it only makes sense "to have a cold". As for a static verb, yes your guess is correct here but "have" is not always a static verb because it is used to signify other actions, eg. to have breakfast, or to have a shower. In these cases "I'm having breakfast/a shower" is perfectly acceptable because you can visualise the action (the basic rule to dynamic verbs). If you say "I'm having a cold" (ie. sickness), then that implies your illness is deliberate and it is welcome, much in the same way one could say "I'm having guests". Hope this helps.

It is more common to say: I have a cold/ I've caught a cold/I've got the flu

I have caught a cold!

You are right. It doesn't matter whether the verb is state or not. Many state verbs can be used in continuous tenses but when we speak about diseases the verb "have" can't be used in the continuous tense. By the way, there is one more mistake. We have A cold.