Doohickie

Yes, yes, yes, to your questions. But trainer miles (and mph) do not equal true cycling miles. Trainer resistance settings are not realistic compared to actually riding a bike out in the world. So comparing the mph on the trainer to the mph you actually ride is not really valid.

Your progress in three weeks is actually very good. Keep at it. The more you ride, the easier it gets and the faster you go. To expand your envelope, try to ride 3 times a week as a minimum. If you only ride on weekends, you lose much of your gains by the following weekend.

As for speed: Don't let it concern you at all. Focus on distance. You can steadily increase your distance by planning your rides out. Eventually the speed will come, but even if it doesn't, don't worry about it. I ride a fair amount and can keep up with social cycling groups no problem. I don't picture myself ever doing serious racing or even hardcore training group rides. So averaging 12-15 mph is plenty fast for me. My wife rides with me sometimes; when she does, we average maybe 7 mph. That's fine too. In fact, rolling at the slower speeds I've noticed a lot more in the neighborhoods than I did at twice that speed, so there are advantages to going slow, too.

Just have fun. With increased riding, the speed will come. If you want to increase your speed, think about joining some group rides in a couple months. Also, you might want to ride some charity rides. Around here they cost $20-50 to sign up, they provide snacks/water along the way, and you can find a group that's running a speed you can keep up with.