The lyrics of the song are basically poetry, and the singer has used some "poetic license".   Some of these phrases are not really spoken in daily life, but the singer has invented these phrases in order to tell the story, and make the song sound nice.   The lyrics are a bit abstract. To some extent, the lyrics are open to interpretation, so there isn't necessarily a correct or incorrect answer :)

The words "between the lines" don't usually mean anything when talking about two people.  But I have read all the lyrics of the song, and in the context of the entire story, it seems like she isn't able to communicate effectively with him, and they are not talking openly and honestly about their feelings.  Instead, she has to "read between the lines", i.e. figure out what he is thinking and how he is feeling, even though he doesn't say it out loud.

"Time to tell me the truth" - she wants to know his actual thoughts, instead of guessing what he is thinking

"To burden your mouth for what you say" - he needs to say it out loud

"No pieces of paper in the way" - she doesn't want to communicate via letters / email / instant messaging / social media;  she wants him to tell the truth, *verbally*.

In this sentence, she is singing metaphorically, not literally.  There isn't necessarily any paper involved - in this situation it means that anything other than honest verbal communication would be unacceptable.  For example, if it was written communication, the paper would be a problem, it would be getting "in the way" of direct communication.

That is my interpretation anyway :)

I think you answered your own question in #1.  It means the way they communicate leave the speaker unsatisfied, guessing and wondering all the time.  Good communication should not leave things to be read between the tines, I suppose.

"No pieces of paper in the way" refers to a similar situation as "between the lines", or maybe even worse  She says "Time to tell the truth / to burden your mouth for what you say", which means to come out with the truth instead of playing games, instead of hiding behind a facade. 

So "No pieces of paper in the way" should be along the same lines.  "paper" usually means legalities that let people off the hook when thing get sour.  If you need to involve any sort of paper in dealing with someone, it means you don't trust him or her fully, that it's less than true love, doesn't it.

Anyway it is thoughtfully written lyrics about relationships.

It reminds me of an old song by Janis Ian with the same title and sentiments.  She also had "Watercolors" (with an incredibly good intro) with a similar theme, and "Tea and Sympathy" (a stunningly sad song), and "Seventeen" (her biggest hit), among others.  The titles themselves suggest what the songs might be like.