Hello!!
1) You don't rommance Sy, she rommances you! (Play the game and you will eventually unlock a scene where she becomes a love interest. In your room, during the morning)
2) That is a known bug. The animation won't play but the intimacy increase is still happening. Just keep spending time with her and you will eventually reach 100%. After that, you can unlock the new scene.
3) There are only two girls in the dating app right now. If the one that loves dogs stopped texting you, maybe you sent the wrong messages and she lost interest. The other girl has only one conversation for now.
4) That is the last part of her story for now.
Thank you so much for your kind words, I am happy the game was of your liking!