Ok, hopefully this will be helpful.
Schooling
The state education system in Catalunya is regarded as the best in Spain, but it is taught predominantly in Catalan (obviously), in addition to this there are absolutely loads of private schools.
Private schooling at Kindergarten level will cost you between 4500 and 6500 a year, plus lunch at school which is often compulsory and will be about another 1500 a year.
We chose to send our daughter to a trilingual school, where they have an equal amount of lessons in English, Spanish and Catalan, but there really are a lot of options here.
Transport
I don't know about Singapore, but the public transport system in Barcelona is fantastic, and very cheap compared to a lot of European cities. The metro compared to London is stupidly cheap, a single journey costs 1.40, but if you buy a ticket for a number of journeys, or monthly or yearly passes it gets much much cheaper.
If you are planning to live outside of the city, then there are very frequent trains into the centre, most going either to Sants or Placa Catalunya, and these are also really inexpensive, but can be quite crowded.
Driving into the city is a pointless exercise unless you have a work allocated parking space, because the cost of parking is really pretty steep.
Property
I can't really help with property prices, because we were looking for something completely different to you. We wanted to live right in the centre of the city because we think that if you are going to live in/near a city like Barcelona then you are better to be right in the heart of it, as there is so much going on with a lot of free cultural events, fiestas in the streets, etc. that if you live outside you miss a lot because you don't feel like getting a train in to go to this fiesta or that free concert.
Try having a look on idealista.com You can change the language to English and it lets you search by area (using the map), and specify number of bedrooms, pool, etc. It is a mixture of agencies and private adverts and is pretty representative of the prices around.
Utilities
We've only been here for a couple of months, so not long enough really to have a clue about average bills for gas, water, etc. So far it seems pretty similar to anywhere else in Europe.
Anything else
As I said, I don't know how much you will spend on property rental, but you should be able to live comfortably on 5k a month, assuming that you aren't going to be going out to the opera or cocktail bars every night (which you aren't).
Hopefully that helps a bit, if you have any more questions then tack them on here, or private message me and I'll see if I can help.