Translation:I have a problem with the secretary.
Día doesn't derive from Greek (that would sound more like "emera" and it would be feminine), but it's a Latin origin, from the masculine "diēs", which got vulgarised to "*dia" while keeping its gender.
There are some patterns, but for some other words you can only memorise. Those original Greek-neutral nouns almost all end on '-ma': problema, dilema, tema, drama, programa, and so on. Then you have the job names ending on '-ta' which you can use for either gender: el/la artista, arquitecta, optimista, poeta, fundamentalista, et cetera. And there are those more-or-less one-ofs, like "el mapa", "el avión", and "la mano".
"Have" is one of the verbs that is used in the progressive tense only in certain situations. You can be "having a drink", "having a party", or "having a shower". These are all uses where it can be expressed with a different verb, "drinking", "partying", or "showering", respectively. But when it comes to using "have" to say that you own something, using the progressive tense and saying "I'm having a car" is very weird. Same with "a problem".
I gave you an up vote for a good answer. However, I suggest that "I'm having a problem" with my secretary is perfectly acceptable.
I suggest that the issue of "ownership" is fairly clear with cars. But not so clear with "problems." For example, I could say "I am having car problems."
Hm, right. Reconsidering this, "I'm having a [thing]" also sounds like you're experiencing something. Like you could "be having a moment", or "having an experience". I think "problem" can fall under that. "I'm having a problem with my secretary" still sounds somewhat weird to me, though. Like you didn't have a choice in choosing one and require a new one, now. "I'm having a problem with Photoshop" is more familiar.