Warm up for working out/other physical activities, but not more running. ~snip~ Plus from what I've heard, it'll be well worth it to be able to do distance running.
OK, I hated running when I first started. Shin splints, improper shoes, 30# I ended up losing...you name it, I sucked at it. You can find my first ever PST score in here somewhere if you want to. It includes my run time.
Now, let me see if I understand your post.
I generally try to run 5 days a week. Unfortunately, the times aren't exactly stellar - I'll generally do it in 2-3 sections at 6mph with 3-5 minute walking in between. Still building up the strength in my legs.
You're running 5 miles a day and trying to run 5 days a week. During those 5 miles/day you're running at a 10min/mile pace and having to walk for 3-5 minutes 2 or 3 times.
I'll address that first. Why not start out with short distances and build from there? You're not doing yourself any good run-walking 5 miles at a 10 min/mile pace. Look up interval/tabata running or Fartlek. That will increase your conditioning much more than what you're currently doing. Nothing is easy at first, but you'll start to see progress.
Goal is to get to the point where i can run for 1-1.5 hours straight
Why in the world would you want to run for 60-90 minutes straight?
and then up the speed steadily until I can do consecutive 8-minute (or better) miles.
Consecutive 8 minute miles sounds like a good 50m target.
Local recruiting office wants 12 min, 30 seconds for 1.5 miles, and I feel it is sort of cheating to just work on my 1.5 mile time only.
Crawl before you walk. Once you have your 1.5 mile time where you want it, then you can start to worry about stretching it out from there.
You have the right idea in trying to train beyond just the minimum standard, but it sounds like you're trying to do too much at once. You have to allow your body some recovery time as well. 5 miles a day (regardless of how slow or fast it is) for 5 days a week is a lot of mileage for someone starting out. I'm not sure if I could run for more than 5 minutes right now. Starting off slow sucks because we all want to be the best at everything as soon as we pick it up. For most of us, that's just not possible. Small bites, baby steps. Assuming you get whatever SOF contract you're lusting after, you'll be training for a marathon, not a 100 yd dash.
You can take the running advice or leave it. I hope it helps. It's all based on my experiences and the research I did when I was training to get my junk kicked in every day.