1. What does 'we're talking away' mean? Is it a common usage?
It just means chatting, although it can also contain the idea of 'with enthusiasm'. It is used colloquially.
2. What does 'to find sb. shying away' mean? Is it a common usage?
In the song I'm not at all sure that 'to find you' and 'shying away are connected. 'Shying away' means 'backing off in fear' much as a horse might do if approached too quickly. It is used but not too often.
3. What does 'take on me' or 'take me on' mean in the context. Is it a proper usage?
'Take me on' can mean 'accept my challenge' or 'employ me'. In this case, it is closer to 'employ me' the speaker wants the boy or girl with whom she/he is talking to accept him/her as a girl/boyfriend. It is used but it is a little unusual in this context.
4. What does 'I'm odds and ends' mean? Is it a proper usage?
In the context of the song this means something like 'worried and confused' but I have never heard it used like this before. 'Odds and ends' usually means' small unimportant things.
5. What does 'to stumble away' mean? Is it a common usage?
Here it means to walk away rather unsteadily as if weak. It is used but not really in common usage in this context
6. What does "It's no better to be safe than sorry" mean? Is it a common usage?
Usually the saying is 'it's better to be safe than sorry' and means 'it's better not to take risks'. In this context the 'risk' would seem to be openly stating your feelings to someone and risking rejection. By adding 'no' the songwriter is saying that not taking this risk is just as bad as taking it because of the negative emotional effect not doing so is having on him/her. With 'no' it is certainly not common useage - without 'no' it is in common usage.
7. What does 'Just a play my worries away' mean? Is it a common usage?
'Is it life or just a play?' A rhetorical question perhaps - reality or unreality. The addition of 'my worries away' is odd. Usually, it is 'take my worries away', however, it is possible to say (without the article 'a') 'play my worries (or troubles) away' - if, for example, the writer is referring to her/his ability to relieve the stress by playing a musical instrument and putting all the pent up emotion into the music. It's not really in comon usage in this context.
I hope that is of help.
Student or Learner