You can walk most places in Barca really easy - and the tourist buses are great.
You dont need a car seat for buses or taxis - I really wouldnt take one unless necessary.
Another tip would be to buy a cheap/light pushchair - it will be much easier when folding up for the bus/taxi.
We bought a m&p aria from ebay, cost about £20 second hand, weighs about 4kg, and also lies flat - it was a god send, we loved it so much that we ended up selling our expensive Loola when we got home and using it all of the time.
It also means that it doesnt matter so much if it gets trashed by the baggage handlers.
You will be able to keep your pushchair until you get on the plane - but have you also considered taking a a sling with you? It would be very handy for busier parts/if you wanted to go on a boat or something - I had a mei tai which I loved, as did ds at that age.
Have a wonderful time - I was in barca last week for the 3rd time - I love it.
Use the tourist buses as much as you can as they are a great way to see the city - the marina is fab, and there is an amazing bar (child friendly) next to the wax works museum down near the harbour - its done out like a fairy grotto.