"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".
と can be used the mean "and" when it connects two nouns. ご飯と肉を食べました (gohan to niku o tabemashita) - I ate rice and meat.
Here we are not connecting two nouns but two sentences. We usually have to use a different grammatical structure to do that. When we want to say "let's do A and then do B", the first verb (A) uses the te-form to tell us that A happens first, and then B. Including the particle と would be incorrect, because the te-form of the verb already implies the word "and".
The te-form is acting like a conjunction to connect the sentences. The first te-form verb will take the tense of the final verb in the sentence.
[invitational] 座ってテレビを見ましょう。(suwatte terebi o mimashou)
The last verb is 見ましょう (let's watch), so the 座って becomes "let's sit down". Let's sit down and watch TV.
[past tense] 座ってテレビを見ました。 (suwatte terebi o mimashita)
The last verb is 見ました (watched), so 座って becomes "sat down". I sat down and watched TV.
[command] 座ってテレビを見て。 (suwatte terebi o mite)
The last verb is 見て (the command "watch!"), so 座って also becomes a command (sit down!). Sit down and watch TV!