No, it's an opinion I've seen around, certainly, but it's definitely not "mainstream".

Sadists can't kill their children (or rather, you can't kill your children out of sadism) for the same reason they can't torture pigs: "human deriving pleasure from bacon" is held to trump "pig desire to continue to live", but that still trumps "human deriving pleasure from causing pain"; and it indicates there's something wrong with you.

The ten month line is arbitrary, but picked because you would have a very hard time making a reasonable case that a baby younger than that is already a person.

I'm not sure what you mean by "why can't we kill babies". Why is it illegal in the real world? I'm sure you can write out as good an answer to that as I.

The bit about discrimination against youths is because I was having a conversation about it a few days ago, prompted by an aside in a a short story I was reading. It's not a common belief, and one I've seen only rarely espoused on LW. The story is "The Right Book", specifically, the line being

He was reminded of all those terrible little signs that said "No more than two school kids in the shop at any one time." Fancy that -- imagine if it said "No more than two women in the shop" or "No more than two Asians in the shop" -- kids were the last group you could treat like second class citizens without being called a bigot.