Learn Spanish (Spain): No estoy pintando, estoy barriendo.

It depends on the ending of the original verb! Pintando comes from pintar (to paint), which is an -ar verb, so we drop the -ar and add -ando. Barriendo comes from barrer (to sweep), which is an -er verb, so we drop the -er and add -iendo.

Where is the word for "I" in this sentence?

It's built right into the verb! Estoy means "I am", so you don't need to add yo (I). In Spanish, it's very common to drop the subject pronoun because the verb form already tells us who is doing the action. You could say yo no estoy pintando, but it sounds more natural without it.

Where do I put the no when using the present continuous?

In Spanish, to make a sentence negative, you always put no directly before the conjugated verb. Here, that is estoy, so it becomes no estoy. You can never split up the two parts of the present continuous by putting a word between estoy and pintando.

Could I use soy instead of estoy here?

No, you must use a form of estar (like estoy) to build the present continuous. We use estar for actions that are happening right now in the moment. Ser (which gives us soy) is for more permanent things, like your identity or profession.