Since my girlfriend and my sister like the neon tetra better, so i will probably go for the neons.
Therefore, I have a few questions in my head now.
1) How long do the neons' lifespan roughly about?
2) I am planning to buy a 20 gal tank. Around how many neons can I put in the tank?
3) With the tank purchase, the AquaClear150 is coming with it. Is it enough?
4) Is a 40 gal pump too strong for the tank?

Im in the uk so dont know much about filtration. The neons only need around 4 times tank turnover and they dont mind a current. For your size tank you can get around 15 fish if you only want neons. Add a few per week so you dont overload the filter.
Will your tank be cycled first as neons like an established tank and dont do well if the tank is under 6 months old.

If it were my tank, I would get 6-9 neons for a school and leave the space for a few more types of fish. 6 is a general minimum and more is always better; any less than that and they may feel insecure or an imbalance of dominance/nipping can occure.

Aquaclears are great filters and a 150 on full power will be enough for your tank. They include a mechanical layer (sponge to filter debris), chemical ( carbon etc which is optional), and biological (usually ceramic tubes that house beneficial bacteria for the cycle). Make sure you have the biological layer because it is very important. You don't need a pump if you've got a good filter (you do). I don't really know much about them, maybe someone else can help.

With good care, neons will live 5-8 years.

I would not cycle your tank with neons. They are very sensitive to ammonia and nitrate, especially when you first get them and they probably will not make it if you try. I recomend cycling fishlessly: http://www.kokosgoldfish.com/FishlessCycle.html