Translation:I have a problem with the secretary.
See this link on Spanish masculine ending in "a". http://spanish.about.com/od/nouns/fl/Words-From-Greek-Ending-in-ma-Often-Masculine.htm
I have given you a lingot because you deserve a reward for that huge streak :)
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.
"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.
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".
Could you supply both interpretations because I'm not quite sure what the second one is, unless you are referring to the fact that English speakers often use the present continuous tense (I'm having) and the present tense (I have) interchangeably.
Eugene Tiffany is right about one thing, though. There is no reason to translate "la" as "una" in the sentence "Tengo una problema con la secretaria."