We travelled by plane ALOT when DS#1 was under the age 2 (free seat) limit.
A couple thoughts:
I totally see your reason for bringing a backpack. I would also bring your sling. This is how I feel safest in airports, and is great for just going around town. Also, if it's a long day, you'll have other family members there (DP,yes?) who can share slinging with you if your shoulder starts to ache. This takes up virtually no room to pack.
A taxi is like a city bus, in that car seats are not required. If you feel more comfortable with a car seat, there's a nice travelling one on the One Step Ahead website, that is very small and lightweight.
I don't know if you've already gotten tix yet, but DC is still young enough to sit on your lap and not need his/her own ticket, so I'd take advantage of that and not bring a car seat!
Things to bring (musts):
Rescue Remedy
Any calming-sleep promoting tincture (we use 'Herbs for Kids' Valerian Super Calm, Nerve Support)
new toys (well, we just go to the thrift and get ones he hasn't seen - this works for long road trips as well)
snacks, snacks, and more snacks. If there's anything that will keep a baby's attention, it's food! Even a 7mo, you can get things to suck on, like teething biscuits, Newman's unsalted pretzels, etc.