Not if the couple were in mutual agreement about the methods of birth control they were using
What if both agreed not to use birth control? The woman shoulders everything and the man gets away Scot free?
If a woman doesn't want an abortion, she should avoid PIV sex.
Why won't you extend this principle to men? If they don't want a baby they should avoid PIV sex
Also, if a woman doesn't want an abortion then she shouldn't get an abortion - not "she shouldn't have sex" 
I'm going to pretend for a moment that the law you so desperately want lilac is in force. Do you think re a married couple who were actively trying, a man should be able to sign the paper and the woman just put up with it? As you seem to be focussing your view on the belief that only men who piss off are men who had one night stands and no one else? Despite people telling you otherwise that husbands do it as well.
And say there were a 'criteria' for a man signing the get-out clause. For example, a "friends with benefits" fling. How would you expect people to prove? What if man says "it's just casual sex" and the woman says "actually we were in a committed relationship and trying for a baby". Or a ONS, the man says they used a condom, the woman says they didn't. Who would you believe? Would you want proof? Would there have to be a court case? That's a good waste of taxpayers money!
As a pp said there'd be 2 classes of children - those officially with dads and those without. Poverty would rise. You've mentioned how taxpayer foots the bill for single parents, this bill would absolutely sky rocket. There'd be more pressures on the NHS. There'd be more children suffering (because despite your smugness about never getting pregnant and blase attitudes to abortion, the fact is babies still would be born wether you think they should or not).
Still waiting to hear how your views are the "opposite of misogyny". That is insanely deluded!
Your arguments are unbelievably ill thought through and you come across as not only a woman hater (yes women can be misogynists too) but really quite dim.
It's at odds to say we want to live in a society that condones 'recreational' sex ( for want of a better term ) and then say to men 'actually it's not like that' because I'm pregnant and you should have considered the very real consiquences of putting your penis inside me
Bollocks is it. Men aren't walking dicks, they all know sex may lead to a baby. Shrugging your shoulders afterwards and saying "but I only had sex for fun" is ridiculous