Learn Romanian: Am o răceală și stau acasă. - I have a cold, so I'm staying home.

și literally means "and". However, in Romanian, just like in casual English, "and" is frequently used to connect two thoughts where the second is the natural result of the first. While you could explicitly say așa că (so), using și is perfectly natural here: "I have a cold and [so] I'm staying home."

Is răceală related to the word for cold temperatures?

Yes! It comes from the adjective rece (cold). The noun răceală specifically means "a cold" (the illness) or a general feeling of coldness. If you want to say "I am catching a cold," you use the related verb a răci.

Could I use rămân instead of stau for "I'm staying"?

You could, but stau is much more natural here. The verb a rămâne emphasizes the act of "remaining" or "staying behind" (for example, when others leave). a sta (which literally means "to sit" or "to stay") is the standard verb for spending time somewhere, like staying home or staying in a hotel.

Why isn't there a preposition like la before acasă?

The word acasă already functions as an adverb of place meaning "at home" or "to home". Because the location or direction is built right into the word, it never takes a preposition like la or în.