The head tube angle affects the steering geometry and by extension the handling.
The seat tube angle mostly affects the seat postion. It made more difference back when all seatposts had about the same setback. Nowadays you can get seatposts with a wide range of setback from zero to ridiculous.
58cm top tube is rather long for a 58cm frame, but not unusual. Since you are new to cycling, you may need more time to acclimatize to leaning forward.
Or, the frame may be too long for your torso length.
I'd suggest you use the
online calculator from competitive cyclist and measure yourself carefully. The "Eddy fit" is a good traditional fit for a level top tube road bike.
If your suggested TT length is close - like within a couple cm, getting a shorter stem to make up the difference is practical. And I agree that those bars look to be pretty long reach. You could replace them with shorter reach bars and it will help a lot.
If it's way off, consider a different bike.
BTW a 58cm bike would typically be just right for someone who is 5'11".