I think to say it's raining one would simply say, "llueve".

Just like in the lessons here, the present progressive of "Está lloviendo" would be "It is raining right now, this very second." and "Llueve." would be "It rains." or "It's raining." If you want to stress the time, use the present progressive one using estar. If you want to say "Llueve hoy" (It's raining today.), then obviously use "llueve". If you want to say, "Está lloviendo ahora." (It is raining now.) Then, you get the point.

darme cuento de que...is that a dequeísmo?

No, it is not, in Spain we would not know, jeje, and actually nobody knows the rule, which is pretty simple actually. Lazarus mentioned it once and it was so simple it stuck:

Just use esto after the sentence, if it works, no que needed.

Me doy cuenta esto. ??

Me doy cuenta de esto.

Estoy segura esto.??

Estoy segura de esto.

Es la primera vez que he visto llover desde abril y es fantástica. Acabo de darme cuenta de que nunca he intentado decir que está lloviendo antes.

To say no he visto la lluvia...sounds like you have not seen rain, sounds weird.

To answer Nicole. The temperature is terrible. It's getting better now, but hace mucho calor probably would be a huge understatement. More like hace (not sure how to say this next part) really freakin' unbearably miseable y caliente. It's getting better now though. down to around 100 durring the day. It was in the upper 130's (farenheit).

You can say either:

Está lloviendo

or

Llueve

Esta lloviendo tiene el sentido de que esta sucediendo ahora, mientras habla

Para me seria correcto decirlo como lo escribio en su pregunta

This is the first time I have seen rain since April and it's great. I just realized I had never tried to say it's raining before.

Es la primera vez que he visto la lluvia desde abril y es fantástica. Acabo de darme cuento de que nunca he probado decir que está lloviendo ahora.

Even close?

darme cuento de que...is that a dequeísmo?

Esta es la primera vez que he visto la lluvia desde abril y es fantastico! Acabo de darme cuenta de que nunca habia tratado de decir que esta lloviendo hasta ahora

That's my best guess