Is Lex Luthor behind it all?

I can see him being behind the CoL. Not sure about the other stuff. At least not directly. BUT, I can definitely see him using the CoL as a manipulative force to turn public opinion against aliens and, in turn, perhaps get himself pardoned as a result, as he was one of the biggest anti-alien forces around.

Once pardoned and released, all bets are off the table.

I can see him being involved in the President and the CoL but, I don't know that he would be connected to the Red Daughter. It seems as though they have the Alien team (Manchester, Kara, J'onn and Brainy) against the Human team (President, CoL and Lena). The human team is a combination of individuals who are afraid of the aliens and who want to either control them or be able to have the ability to defeat them. The alien team, of course, is trying to prevent that from happening and in the case of Manchester, perhaps even take the fight to them. I see the Red Daughter being someone that the human team would be against. I also believe Lex would be on that Human team - against aliens. Plus, I don't know that he would want to encourage training an alien into being a weapon that could be used against humans in the US.

On a second watch, I noticed that I missed the Russian saying, "Our friend is working on it," when asked if the Red Daughter could be controlled. This does make it seem as though there is an outside source, perhaps Lex, that is involved in that situation. If Lex can control the Red Daughter and then use her to attack the US and cause a conflict between Supergirl and Red Daughter that damages the infrastructure of the country or between the US and Russia itself, then that would possibly be the excuse to demand that more humans get superpowers.

So, yeah, he could be "pulling the strings" on all of it.

It wouldn't surprise me if this has Lex's fingerprints all over it.

And as for crossover part 3, that line about gender...I would've thrown the penalty flag!