Kat, I wouldn't even think about speed yet. If you are comfortable at 5.5-ish range, then just stay right there. I would first work on getting all the way to a 5k at WHATEVER speed is comfortable. It will likely fluctuate between 5.0 - 5.5. And that is a great tip when you are working to run the whole way through. It is almost like intervals, just running intervals instead of run/walk. For example, you might be running along at 5.5 and start to feel fatigued. Instead of going to a walk, just drop down to a 5.0 for a couple of minutes until you feel refreshed, and then go back up to the 5.5. But that won't be for a while yet, so no rush there.
And I know it is tough for you with both weather and schedule, but I would recommend getting outside for as many of your runs as possible. For starters, any race you do is going to be outside, so it is always better to train that way. Secondly, speed varies WILDLY from what you can do on a TM to what you can do outside.
The trend tends to go like this--as a new/beginning runner, you are faster on the TM than you can go outside. This is because outside requires more effort and exertion. But once you are comfortable running outside and have gotten your pace back to where it was, when you switch to a TM, you go slower than you do outside. This is because you aren't as "free" on a TM to get your full, natural stride in, so you naturally have shorter strides.
But as with anything, it is highly individualized--these are just the more normal patterns.
For me, my first 3 months of running were 98% on TM. OMG, I was terrified of moving outside. But eventually I did. And now it is the opposite. I literally NEVER run on the TM. If the weather is so bad out that I can't run outside, I just switch my running days. Or suck it up. Running outside is a million times easier for me. I honestly do know if I could even run 5 miles on the TM right now. I doubt it. But when outside, 5 miles is my distance of choice...not too short, but also not so long that I get bored.
Sounds like you have some great goals. I have some, too, and that helps. My first is to get back to running regularly without pain! I am trying not to focus on time or anything. Just good form and no pain. Once I am completely healed, I will begin training to run a 5k at a 9:30 pace and then at a sub-9 pace. I have already done a 10k at a 9:50 pace, so I am hoping that these goals aren't too lofty and can be achieved by September. I am not sure how long I need to work to get to those speeds. Like, I know how many weeks it takes to train for a Half or a Full. But no idea how to plan for increasing speed.
Then my long term goal is to do the Princess again next year (or maybe the Donald in January), but run it to PR. And then once I have a good (for me) PR in the Half distance, every other Half I ever do from then on will just be for fun and with friends and stopping for pics, etc. I just need that one really strong time.