As a flight attendant for thirteen years with two companies, I learned a lot about traveling with babies by just watching what worked and what didn't with passengers, but the real lesson was ahead of me. Now I'm usually flying alone with my own three between Europe, where I now live, and California, where I'm originally from about every six months. We've also taken quite a few flights within Europe, the Middle East and domestic flights in the States. I've actually lost count of how many, and now we can claim we’ve done stand-by, full fare, low cost, charter, etc.
I was contacted by someone claiming to work for a company who said certain car seats, were allowed on their airline, forward facing, in aisle seats. They had decided that some seats had a less obstructive design and therefore, it was allowed. I couldn't confirm this and I would suggest talking to the crew if you are interested in doing this.
Take snacks and again, don't worry about security. I've had no problems myself and no reports. The FAA requires that the food be in a ready-to-eat state. For example, you can have jam on a sandwich but you can't bring on board a jar of jam. The snack should be in whatever state that it needs to be to be simply taken out and eaten.
Liquidy foods might be subject to more screening. This would probably be the explosive test strip, which is quick and doesn't come into contact with the food itself. But I don't recommend soups in principal, just because of the messiness and awkwardness of feeding them to a child on a bumpy flight.
I get mixed reports about jellos and puddings from home. Now, many are conveniently packaged in squeezable pouches, which not only eliminate the security issue but avoid the need for a spoon.
Remember that in the worst-case scenario, security will simply remove the offending item without much comment except for explaining why. You won't hold up the line and you won't be subjected to any fines or lectures.
You might want to avoid bringing snacks with peanuts in them even if your child is not allergic to them. Many airlines have elimiated them and people with peanut allergies can be so sensitive that even having nuts around them can set off a reaction. This could be a scary thing for a child to witness, not to meantion the guilt you would feel. Yes, this is highly unlikely (and those with this allergy usually bring supplies for their condition) but why take that risk? Some children love peanut butter and it's convenient. The problem seems to be more peanut products that release dust into the air so maybe use this as a guide. You can simply just ask those in your area if it's alright, if you're comfortable doing that. Do what works for you. Also, peanuts in nut form are toddler choking hazards. All the Flight Attendants know the Heimlich maneuvre but none of them need to practice on your child. Grapes are another choking hazard you'll find on flights. If you bring them from home, halve them (even for older toddlers) and look out for them with your meals.
You are allowed to bring liquids meant for the baby in "reasonable" quantities. The problem hasn't been the quantity but the age of the child. There isn't a stated age but be aware that once your child is walking, that apple juice allowed in just a few months ago, might be taken away this time.
I love my hand luggage scale and recommend getting one. I can't say which brand is best. Try it out at home to double check that it's correct. Have an adult stand with the bag on your bathroom scale, subtract the person's weight and check that against what the hand scale says. Then pack it along with an extra battery (mine decided to conk out on the day of a transatlantic so don't make my mistake!)
Try not to "all go in one bag" because these are the people airlines make a tidy profit from. Also, considering ordering toiletries on line and having them sent to the hotel or whomever you're visiting. Be sure to clear this with the hotel or your hosts first. Toiletries really drag down the weight and you're guaranteed to have the correct diaper size than if your mom tries to figure it out at CVS...
Depending on the length of the trip, and your airlines carry-on allowance, you might want to bring an extra bag with extra supplies. The backpack could go at your feet during the flight and you could pop in the overhead a duffle with the extra diapers, changes of clothes, extra formula, etc. Basically, what you're not likely to need during the flight unless there's a problem (delay, spill, etc.) This is especially good if more than one adult is traveling. Lap babies often allowed an extra bag.
If connecting, you might want to avoid high-top laced shoes since you might have to remove them a couple of times in security, although I don't think it necessary to pick your footwear specifically for this purpose as the TSA recommends. I wear what is the most comfortable, even if I need an extra second to lace them up after security.
Children, at least in the U.S., don't have to remove footwear anymore. The age limit is 12 but my very tall 12 year old took hers' off anyway, to avoid being challenged.
You may be concerned with a very expensive stroller. It would be easy for me to say to simply not bring it but many parents want their child to be comfortable in the familiar stroller at their destination. You might want to research what would happen if it were damaged. Would your home owner's insurance cover it? Is there a bag that the manufacturers recommend? Something specific to your model? Is it still under warranty? Do your homework before deciding.
Some toys are not allowed in the cabin. No toys with remote controls are allowed to be brought on board. The toy itself can be brought onboard but I recommend putting the remote in the checked bags. There is also the obvious rule against any toy that looks like a weapon. No toy can contain more than 3 oz. of water...but on a practical note, I wouldn't recommend bringing any toy with any water at all onboard, as amusing an idea it might be!
When my son was small, he loved anything with wheels. I'd bring a package of small cars, trucks or airplanes. Plastic animals were also winners with my toddlers. When the girls got older, and past that magical age 3 when it seems all toys are "allowed", those small plastic dolls with rubber clothes kept them occupied. All three of them love small animals.
Other suggestions have flooded into me. Pinwheels, pipecleaners, nesting and hooking toys (like blocks that fit into each other), finger puppets, playdough, etc. have all made flights go faster with little ones. Many of these items come with warning labels, not recommending them for under 3's. I found this very restrictive as mine approached this age. On flights, I was right there next to my child, not in the kitchen making dinner or doing laundry, distracted and letting them play alone with what is usually a choking hazard. I would push this limit to a reasonable degree for travel toys, just because of the circumstance and the restrictions on toys I was already dealing with on an aircraft. I'm not promoting buying dangerous or age-inappropriate toys but do what works for you.
Don't go overboard. Keep it simple and easy. Don't bring a truckload of stuff they'll hardly notice and you'll have to lug around. I've brought too much and I'm not alone in making this mistake.
Often, parents have recommended "nighttime" diapers for long haul flights, to reduce the number of times changes are needed. I didn't do this for a few reasons. Those diapers were among the many that gave my son a rash and they didn't seem to fit as well. Obviously, these may not be your problems and nighttime diapers may be a very good option in your case, especially if your child is already using them at night.