If he was born after 2006, then yes, he can have a british passport. However, apart from the birth certificate, the embassy will require other documentation - the hospital bills for ante-natal treatment and birth will be handy, as will photos of his mum when she was pregnant. If the embassy are not satisfied with the documentation supplied, they may request a DNA test to prove paternity.
Oh, and it doesn't have to be 'when he is older'. The sooner you apply, the better.
[I think we should be okay - Ruby met the Consul from Manila last week and the pregnancy is very obvious. She suggested that I mention the meeting when I submit the passport application.]
If he was born in 2006, or earlier, and you're not married, then things will be rather more difficult.
With regard to your other question, yes you can obtain a 38 day extension to your initial 21 day entry. The cost is 3,020 pesos at the Bureau of Immigration (IIRC, that's cheaper than obtaining the visa in London).