I would call and start asking specific questions. We've flown a lot and I'd have to say they are all similar on infant policies.
Find the cheapest ticket for you and then go from there. 2 years and under can be on lap for most airlines (not safe at all but cheaper for sure); most will give you a row with a vacant seat next to you if available (typically on smaller flights and/or at less popular times). I've always been placed next to other parents with children too. If you do get a seat for the dd, some give 1/2 off prices; like I said, since you're looking for inexpensive, find the cheapest for you and ask specific questions so you are prepared and can make a good decision.
Good luck! Hope my rambling helped