You need to make sure Sparky is shut behind a door before letting Lexi out. Do try to follow my integration steps exactly and precisely. Each time there is a negative interaction, things get set back even more. You must make so that there is no possible chance for it to occur. Just because Sparky was not in the room in view at the time, he was fully free and able to wander in at any moment, and you could not possibly know at which moment he might come into the room where you and Lexi were. You need to make it so there never is any possible chance for Lexi to get cornered.
I am wondering how it is that you say Sparky hasn't seen Lexi in a week, but 2 days ago you said he jumped the gate and was in her room. So he certainly could have seen her then.
It is fine and good for him to have brief peeps at her with NO possible way, in any manner, for there to be any physical contact. Really try hard to follow Step 3, Visual Familiarization-
http://wvcats.com/integratingcatspage2.html#visual . Maybe switch Lexi's safe room to the 2nd bedroom- the only room in your house with a door (not counting the bathroom). Then you could manage to do "peeps through the crack in the door". Sparky should be at the Visual Familiarization stage of the integration process.
If Lexi is still staying in your bedroom with the baby gates in the doorway, and she can fit through the gate, how are you keeping her from coming and going at her will?
I am thinking that since you will be taking in two more cats in the near future, it would be best to rehome Lexi. She sounds highly adoptable. I know you love her, but love sometimes means letting go. It isn't going to work out for you to add two more cats. Every time a new cat is added, it adds more stress to all the cats, and changes the relationships and balance the current cats had. There is also the expense for vet care for all the cats, which you are not really able to afford. Plus, you are not really set up for proper cat integrations.
At some point, things really do get to be too much. We cat lovers have to say "no" at some point, no matter how much we want to say "yes". It can really get out hand and become a bad situation for all the cats and the humans if one is not very careful. That is why it is best to adopt out any cats one can.
Regarding a cat flap in a window, you can cut a board the width of the window and a little more than the height of the purchased cat flap. Then shut the window and screen on top of the board. Use some sort of hardware gizmo the affix the board to the window- some sort of rectangle thingy where one end gets screwed to the board, and the top to the window.