I've got a jogging stroller and it's great! I can put my 2 yr old girl in it, and she'll eventually fall asleep while I get a nice workout.
I have a BOB Revolution, but it's a bit expensive. Jogging strollers either have swivel-front wheel (that can lock) or fixed-front wheel. The swivel is good for maneuvering around turns/corners and walking. If you have a fixed front wheel you will have to lift the front of the stroller when you make a turn, but the fixed-front wheels are great for runners.
Most jogging strollers uses air tires. The bigger wheels (16" diameter rear wheels) will offer a more comfortable ride for your child. I think they come in 8"-12" wheels
Jogging strollers offer attachments if you want to hook an infant seat carrier onto it.
Jogging strollers are not very 'compact', so make sure you try folding the demo and making sure it can fit in your trunk. They're not as lightweight either.
Look for a 5-pt harness for your child, and also a safety-strap for a brake (in case you have an emergency and need to stop quickly). I think most jogging strollers have these features.
Then the amenities (like cup holder) may be accessories that you need to purchase separately if you want one.
I think jogging strollers are rated to hold a child between 50-70lbs. so you'll get alot of 'mileage' out of it.
Good luck w/ your search.
I have no idea about bike seat attachments, but would recommend a helmet for your child!