"Let's sit down and watch TV."
据わって or suwatte is the te form of the verb Suwaru which means "to sit". The te form is used in progressive forms and so it is used here as an action that links to the verb after it. Using the te forn this way doesn't affect the tense of the whole sentence, which is determined by the last verb or action, so multiple te form verbs can be used to link a number of actions. The last verb is in the volitional form, so "let's" is the "-mashou" part, and the te form verb is why there's an "and".
It's similar, but the "~tari ~tari" form implies that you plan to do other things as well. "Let's do things like sit down and watch TV."
Because it means sitting and then watching TV in distinct sequence rather than simultaneously