It may be worth remembering that people mature at different rates. Comparing yourself with others your age is not a great comparison so don't run yourself down! Furthermore you do not have to be a runner to coach runners, so just because your Dad is 'only average' does not mean that he does not know how to coach.
The fact that you are running regularly is going to help you a lot and will pay off in the long term. The first thing to do is keep a diary of your training with relevant details like how far, how fast, the weather, how you feel (good, bad, tired, motivated etc.) and try and be consistent (which you appear to be!).
Joining a running club is a great idea, but it takes a bit of courage to turn up not knowing anyone or what standard is expected of you. But it would be a good move!
But as cougie says keep the intensity down, try to be consistent with the amount of training you do each week and go for it at the parkrun! The parkrun would be a good place to find out about local clubs and which one would suit you.
Most important go out, run, enjoy yourself and the improvement will happen
I have to say I have no idea how diabetes and an underactive thyroid combine. The endocrine system is complex as various organs work together (or don't with diabetes).
However taking thyroxine is likely to make you feel a lot better! (but not necessarily!) The thyroid normally works with your hypothalamus to regulate your body temperature so the medication should increase your body temperature. A higher body temperature should help you to recover from training quicker, eat more and generally improve your digestive system. So less constipation! But this is all 'generally' because it will affect different people in different ways.
The 'doping' side of it is controversial, but you would have to be a moron to 'dope' without medical assistance. Too much thyroxine is likely to make you ill. You (someone who needs it) are only artificially providing your body what it should have anyway. You certainly would not stop taking insulin because it is 'doping' although it IS on the banned list! (it is used to promote hunger in endurance athletes to help them eat enough calories!).
You say you have no symptoms but what about body temperature? I would suggest buying a good quality 'tympanic' thermometer (a in your ear one). I use a 'Welch Allyn Braun 4000' (second hand on ebay they are £40 ish) your body temperature should be 36.5C. A low temperature is the main symptom (but you don't have to have one!). I know of people who were 34.5C! If you take thyroxine you should keep your temperature below 36.8C or otherwise keep your levels checked, hence the 6 month blood test! Worth noting how your temperature changes.
There are a LOT of books out there but most are peddling an expensive supplement regime or a certain type of trademarked medication so beware! Your Dr may be able to suggest one that is helpful if you want to learn more.
In short, how the medication will affect you is difficult to predict, but likely to be positive! So don't worry. And were doping is concerned, thyroxine is not on the banned list so don't worry!