Hello, congratulations on starting to run! I'm further along the same programme as you and loving it too - it's great to feel an improvement. I've done quite a lot of research into the best way to stick to the programme and create a solid basis of fitness.
With that in mind, I would really caution you against running every day. The way that getting fit works is that if you run when you haven't before, you get lots of microscopic tears in your muscles. These then heal, and once they've healed, they're stronger than they were before. Then you run again, tear them again, and they heal again even stronger. You keep repeating this pattern until you have muscles that can withstand much more activitiy. But when you're just starting out, if you don't give yourself rest days, even if you feel fine your muscles won't be getting the time they need to heal up properly. So if you run every day, they just get more and more torn until eventually you get a serious muscle injury and you won't be able to run at all for a while. So please, take your rest days! It's for the good of your fitness. And then, once you've got a lot stronger a few weeks down the line, your muscles will be able to take the strain of running more often. I hope that makes sense - I know a lot of people who were really enthusiastic and they ran every day when they were starting out, and then they got injured out of the blue and had to stop. And there's nothing more discouraging than that, so it's best avoided!
As for water, you shouldn't need to take a drink if you're out running for less than an hour - just make sure you have a drink when you get back.
I don't see much problem with skipping the initial walks if you feel you can! Just make sure you don't overdo it and get discouraged, I think that's the biggest danger. Take it slow and steady, you'll get there. Good luck and look forward to hearing about your progress!
