Well, I completed this last night and was delighted to do so.
I've noticed that when I start a run, my legs are OK but I get out of breath. By the second or third run it's the other way around - breathing is OK but legs get more heavy. Then when I finish and do my warm down I can feel my hamstrings tightening. Is there any benefit to be had from compression clothing which might help with this? Or are they just a gimmick?
Compression clothing is not a gimmick, although widely misunderstood, hence why people think they dont work. They are not a miracle cure for muscle aches nor would they really benefit you in this instance. Most probably what is happening is your hamstrings are just still not adapted to the demands of the running, which are increasing as they struggle to keep up. Doing good hastrings stretches will help: take a couple of days a week to do a session of mobiity work, rather than the lipservice 'post run stretches'.
Google Hamstring stretches or better still sign up to something like ROMWod or MobilityWod
well done! you seem to be finding your rhythms...they will fall together soon and your legs and breathing will synchronize - which will make it much easier 
Make sure you do your stretches, and try and add some body strength training sessions into your week - they will help your muscles grow and adapt (pretty much as Rignold suggested...sound advice) Good luck!
It’s all in the calves they say 😃
Just go slowly and you’ll be ok
I like to do a calf raise with a heel drop. Nice and slowly, up,then down. Bliss
Take care 😃
Cheers guys
As Rig says, get some stretches in...regularly .. rest day exercise is essential... it is early days yet and everything needs to build up...your body is still wondering what on earth you are doing to it
Week 1 run 3
Week 4, Run 1
Week 4 Run 1 😃
Week 4 Run 1, nightmare. 
week 8 run 1
