Isn't teanga a feminine noun?
Yes, but DNTLS DTS stops the lenition.
Are these letters referring to the initial consonants of the word not to be lenited? What do the final three DTS stand for?
A word starting with D, T or S isn't lenited after a word ending in D, N, T, L or S.
Including after "an"?
Yes, an is a word than ends in D, N, T, L or S, so you have an deirfiúr, an teilifís and an sráid, even though they are feminine nouns.
It's if DNTLS is followed by DTS
Can the second "si" be missed in Irish as in English when you could say "she reads the dictionary and  speaks the language"?
No. In English, the subject comes before the verb, so the first "she" is still, technically, before the 2nd verb. In Irish, the subject comes after the verb, and if you left out the 2nd sí, the 2nd verb wouldn't have a subject.