The key thing to keep in mind is that lips touch makes "lips" the subject and "touch" the verb. On the other hand, lips that touch is a noun phrase. that touch acts as an adjective in this case.
Lips that touch mine
is short for
The lips that touch my lips
It is a noun phrase and not a sentence. On the other hand,
Lips touch mine
as it stands is a sentence that informs the reader that lips, in general, touch yours. It's not really interchangeable with the noun phrase above. You could make it specifically refer to some lips that were previously discussed
The lips touch mine
but this is still a sentence.
Your last question seems to ask if
Has the same meaning. This is another sentence that just indicates that touching is something lips do.