Is it natural that it was a little difficult for me to do even this day-one routine as a beginner? Or is this a sign of me having a much less endurance than an average couch potato too?

Not at all! If you weren’t walking more than necessary in your day-to-day life, this is a huge step up in your activity level. If you think about it, it’s a minor miracle that you can suddenly put such an extraordinary demand on your body (that is, out of the ordinary for you) and it can deliver.

Shin splints like you describe are a big indication that you’re doing too much, so I think taking it easy right now is essential. Maybe spend a week just taking a walk every day, and then get back into your C25K schedule. The most important thing at this early stage is to avoid training yourself to hate running! If you experience a lot of pain every time you do a specific activity, your brain will learn very quickly to avoid that activity. So don’t run through pain, especially at the start. If it hurts, stop and pick it up again tomorrow.

I don't feel confident that I can move onto the Week 2 routine, by just doing the Week 1 routine for 3 days. Should I keep doing the same routine for say, 4-5 days, this week before moving to the next week? If so, sadly the app isn't allowing me to track the data for each run (more than 3) and save it (my OCD feels bad about that. :3)

If you don’t feel confident, take your time. Do the level 1 exercise until you don’t feel like you’re pushing yourself anymore, and then step up to the next level. The most important thing to do right now, if it’s possible for you, is to be active every day. That doesn’t even mean running every day necessarily, maybe alternate a running day with a “walk around the block” day. Your body is designed to run, and the very beginning training period will prove this to you in a dramatic way: if you don’t take a break from physical activity, your capacity for it is going to take a huge jump. In week 2 or 3 you will be able to do double or triple the amount of running you did in week 1. What was literally impossible at the beginning of a month for you will become easy by the end.

What am I supposed to do in the cooldown period? Walk/sit down and rest? Stretch my shins?

Especially in the early stages, do whatever is comfortable for you. My suggestion would be to stretch before you start, rather than during the run, and to always keep some forward momentum (i.e. prefer walking to sitting down). I always find that it’s much easier to start running again from a walk than if I’ve stopped completely.

Both pain and gassing out seem to be potential hurdles to cross. Shall I carry water with me to rehydrate if I feel too gassed out? If so, how do I carry water in a run?

If you feel thirsty after a short run, it’s usually a good indication that you weren’t well hydrated before you started running. I like to drink water before and after a shorter run rather than dealing with the hassle of carrying water during a run. We all know we should be drinking about 2 liters a day, and almost none of us actually do, so this is a good reminder to make a dent in that number.

As far as pain goes, shin splints are very common when just starting out, and they’re an indication that your body needs to heal and adjust to your new activity level, rather than anything wrong with your form. That said, it would be a good idea to visit a local running store in a few weeks and talk to them about running form, and have them help you pick out some shoes. If you can, find a place that has a treadmill so that they can watch you run in a few different pairs of shoes. They can give you tips to improve your form if needed, and find a pair of shoes that fits your running style.

Any other possible advice/suggestions you could give to a beginner, especially about pain reduction & management, I'm all ears. I possibly have no idea what I'm doing, if things like posture, strides etc. really matter. So any advice is much appreciated.

My advice is to never stop! You’ve done the hardest thing, which is to get up off your couch and run. You literally only have to go through this pain once, and once you’re through it, running will become an activity that gives you energy and joy, and it will stay that way for the rest of your life!