I absolutely agree with AJock.
Ignore the heart rate monitor for a few weeks, all it seems to be doing is making you anxious.
I'm back into running again and never use a monitor, ever. It's too much to be thinking about for me. I set myself a goal and work on achieving it, while paying attention to how I feel. I'm using the treadmill too.
What I'm doing is gradually increasing my mileage. I love Jeff Galloway's philosophy, he's such an enabling runner and coach. So, following his guide, I run for 2 mins, then walk for 2, and repeat this 15 times for a workout. I do this three times per week, then next week, I run 3 mins, and walk for 3 and repeat 12 times. And so on. I keep getting to 7 min miles and then being interrupted by life events that force me to take a few weeks break! Arggh! But, I'm back at it again and determined to keep going until I am running 30 mins+ without stop.
In addition to the goal to increase mileage and time I can run without stopping, I also work on increasing my speed. I can powerwalk at 5.6 mph, so I start my running speeds at 6.5. Each 2/3/4/5/6/7 mins I run, I increase the speed of the next running interval by another +1 mph. So, after 5 running intervals I am up at 7.0 mph and loving it.
Sure, my heart rate is shooting up, but by gradually increasing the distance of the running interval each week, my heart is conditioning itself without me having to worry about it. I focus on my breathing at the higher speeds: in for 2, out for 2, and I think enabling thoughts: "I can do this," "I'm getting so strong," etc, and then there is no room for anxiety.
I usually end up, by the time I get to 7 min long running intervals, which I do 10 times, at a total workout length of 8+ miles, and I do this 3 times per week. I am never worried about my heart rate because I know that I can do this because I am gradually increasing the demands I place upon my cardiovascular system and body overall.
Cardiovascular conditioning improves with slow, gradual increasing of the workload and demands you make upon it.
So, for the last 5 running intervals I return my focus to speed. I may continue the gradual speed per interval (i.e. interval 6 at 7.1 mph, interval 7 at 7.2 mph, etc) or I may keep increasing the speed of each successive minute within the 6 mins the interval lasts (min 1 at 7.1, min 2 at 7.2, etc) and usually, I do a combo of both of these approaches. By the end om my workout I can be running flat out at 8 mph. Yes, it's hard and I'm blowing out air hard. But, I maintain focus, I keep the breathing steady, even if faster paced.
I may be feeling very fatigued by the end, and maybe slightly pukey, but the point is: I know what my body is capable of, and I know that I have to push it harder to see the improvement I want and I'm fully prepared to do so because I want it. There's no point in worrying about it. You have to learn to trust yourself and trust your body. You will have to push harder and force yourself out of the comfort zone if you want to improve cardiovascular conditioning and run faster and further. Until you try doing it, you won't ever be able to do it. Your body always responds to the demands you place on it, either by increasing muscular strength because you force it to lift heavier weights, or by improving oxygen consumption and deliverance of glucose and oxygen to the muscles becasue you forced your body to run faster during today's workout. The body responds "after the fact" as it were. You won't suddenly be able to run comfortably at 6.5 mph, if all you have been doing is running at 6 mph. You will have to run a few laps of the treadmill at the faster pace, and then repeat this two more times in the same week, before your body will respond with improved conditioning, which you will experience as "hey, this feels easier to do this week," when you repeat it the second week. Then, by the third week, you have to increase the challenge again: either you run further or you run longer.
Rest is crucial. I used to think I needed to run every day when I was younger (I was a runner as a teenager and at college), but all I got was injured every 6 mths. Now, I see great improvement on three running workouts a week. You don't need to run more than this, and in fact, if you are trying to increase your running speed and/or distance, cutting back to three running workouts a week is adviseable.
But: you will have to step out of the comfort zone, mix it up, play around with speeds, play around with distances and increase the challenge each week to continue to see the cardiovascular conditioning you want to achieve, at whatever speeds you decide to run.
Notes on your post after re-read: why do you not expect your heart rate to increase when you increase the speed at which you run?!??! Of course it will. Temporarily, you will have to accept this. As you repeat running at the higher speed, and your heart responds by getting stronger, the heart rate will lower at the same speed. This is biology. And yes, running outside is more of a challenge than running on the treadmill, since there's no moving floor beneath to assist you moving right along! So, yes, your heart rate went up and you perceived this as more difficult. But, in order to see improvement you will have to put up with some initial discomfort as your body responds by getting stronger. Not pain, but discomfort, yes. I know that the first workout of each week, the one where I do the new distance for the first time, is going to be less confortable than last week's last workout. And it is. I acccept that, acknowledge it as it happens, grit my teeth and run through it.
Go to runnersworld.com and see tips on how to train for that 5k, OK?
Hope this helps,
Clare