1. What's the weather like today?

2. How is the weather today?
3. Whhat's today's forecast?
4. How is it outside?
(Are they all interchangeable? are they all the same and grammatical?)

5 answers

#3 is different, because it involves making a prediction. The others just require an observation.

1,2,4 are all the same, and colloquially acceptable, though maybe not strictly grammatical.

#1 follows the rules of syntax.

I guess, #1,2 could also be interpreted as observing the current conditions and extrapolating to the rest of the day.
3. Correct the spelling of "what's."

For most part, these mean the same thing, although 3 has a different meaning. In the other sentences, you ask what the weather looks like on that day (perhaps to another person). In number 3, you ask about the prediction of the weather for that day.
Hmmm. "what's" is a valid contraction for "what is"

what's your problem?

:-)
There are two hs in 3.
oops - my bad. Just read the first line.