It’s a classic case of Kane confusing love with ownership. You’re right that it seems counterintuitive—if he loves her, why make her miserable? But for Kane, Susan wasn’t just a wife; she was a trophy he had to polish to prove he was right.
Here’s the breakdown of those two head-scratchers:

1. The Pressure to Work
   The pressure wasn’t about Susan’s career; it was about Kane’s reputation.
   The Scandal: Kane’s political career was destroyed because of his "love nest" with Susan. To the public, she was just a "singer" from the wrong side of the tracks.
   The Vindication: Kane believed that if he could force the world to acknowledge her as a Great Opera Star, it would prove he didn't blow his life on a "chippie." If she was a "Great Artiste," then his sacrifice was noble rather than tawdry.
   The Power Trip: When she begged to quit, he famously told her, "I'm the one who has to be convinced." He wasn't listening to her heart; he was fighting a war against the public's opinion.
2. The "Prison" of Xanadu
   You’d think a trip to NYC would fix the boredom, but Kane’s ego couldn't handle it.
   Total Control: In New York, Susan could meet people, hear gossip, and realize how much the world laughed at them. In the "castle," Kane was the absolute monarch. He kept her there so he could be her entire world.
   The Walls of Spite: Xanadu was built as a monument to his own isolation. After being rejected by the voters and "betrayed" by his friends, he retreated into a fortress where he could surround himself with statues that couldn't talk back.
   Collecting People: Just like the crates of art, Susan became a possession to be stored. Taking her to New York would mean "sharing" her or risking her leaving him—something his abandonment issues from childhood wouldn't allow.
   Kane’s tragedy is that he thought providing for Susan (the opera house, the castle) was the same as loving her

2 - It seems to me he just got very bitter after the opera thing didn't work and didn't really want any human contact, thus staying in the castle all the time. Note that even in the picknick(?) he was holding we never see him among the guests, only in his tent.

Early on, Susan said (maybe half-heartedly) that she wanted to sing. So Kane tried his best to make her a singer, supposedly to show his love for her. But his overzealousness got to the point that she resented it. Finally, Susan realized that he did this not to show his love, but to feed his own ego. He didn't want her to leave Xanadu because he basically saw her as an object at that point, the same way a king doesn't want his concubines to leave his castle.

Yeah, it was a challenge for him. But one born out of pettiness.
So many stories, so little time.

wow, I didn't even think about it from that angle. I just saw that helping her make it as a singer was yet another challenge for him, I'll have to watch it again.

my understanding is that her singing career is a reflection on him.
Correct. They explain it in the movie. He's trying to remove the quotation marks around "singer" in the headline that exposed his affair. If she's a real artist, his affair looks less sordid. Otherwise, it's like he left Emily for a mere showgirl.

for the first point, my understanding is that her singing career is a reflection on him. if she quits, it makes him look bad.
not sure about the second one