I completed c25k in the past and it worked great. Then I quit running and I'm starting over now. I just finished week 5.
You can do this. The weather won't stay hot forever, it just seems like it. You've got good goals and a good plan. It's hard to always get your rest with a new baby in the house, just do the best you can.
Something that really helped me was to ignore the time table. It's an 8 (or 9 or something?) week program, but it's okay if you don't finish it right on the button. I have had to repeat a run or take an extra rest day here and there and I don't think anything of it. I'll finish it when I finish it and my body will let me know when that's going to be.
Best wishes to you,