how annoying about your neighbours arcadia
TV on till 2am is inconsiderate - and if you can hear it, it is too loud, regardless of how thin the walls are.
Am so sorry this is adding to your stress. I totally understand what you mean about anticipaing an annoying noise - I used to find that a lot with DH's "clicking" - the moment I heard just one click my blood would start boiling!!! Also finding it with annoying teenage scooter boys, zipping up and down the road, but they seem to have given it a rest the last couple of nights.
You should definitely try to approach them about it, I know it would be difficult finding the words. If they seem nice enough, they might be responsive... you never know, they could be totally mortified if they realised they were disturbing you! Just say something like "hey, hope you don't mind me mentioning but I'm starting back at work soon and need to get some early nights...."
Yes, I believe the medication can help you sleep through noise (especially at the beginning - I was totally zonked out!) and over time it will also reduce your anxiety.
Would DP be supportive though, in terms of getting up for DD if she wakes? I don't think anything would have woken me, the first few weeks of taking mirtazapine - DH got up with her every time, although thankfully those times were few.