If you are unfit, slightly overweight and have not cycled for 7 years, you will be winded. You are not used to this form of exercise and as you state, the bike is not really suitable for you and probably quite heavy as well.
Don't worry about heat rates and zones - that is not what you are after. You are not training for a race, you are trying to get fitter and enjoy yourself at the same time. It should not be a chore and it should be enjoyable.
Get used to the distance you are doing right now and keep repeating it.
Listen to your body. You will know when you are good to extend it.
It may take you are few rides, it may take you a few weeks, but don't extend it until you feel ready and don't worry about how long it take. Everyone is different.
I also know the challenge of returning to cycling after only 4 months off the bike, but I have had back surgery after a disc ruptured (5 months ago) leaving me with chronic pain and what is looking like permanent paralysis.
I struggled with my very first ride back on the trike (I can no longer balance, or more accurately, no longer have the ability to stop without falling over when I do because neither leg will hold me!). I barely managed 4 miles as well and it exhausts me physically. It has taken me 4 weeks getting out roughly 6 times a week, but I can now manage 9 miles and have done one 10 mile ride which confirmed it was too much for me!
You are going to have to listen to your body. Once you have some miles under your belt, you will find the miles get easier and the weight should come off, if you don't eat too much when you get home again! It is always the temptation! I'm now up to 140 miles on my trike and it is getting easier slowly.