

Another thing to add on to why weight isn’t a great measurement - You will put on muscle as you lose fat, especially if you shift your diet while you workout. It’s fully possible that you lose 5 lbs of fat in a month while putting on 5 lbs of muscle. Both are great strides forward, but the scale will show no change.
If you are going to use a scale, get one that will do body fat analysis as well. They aren’t 100% accurate, but its another measurement that gives a closer look at your actual fitness.
Belly fat is unfortunately just calorie reduction. You can’t target fat on specific parts of the body, just overall.
Improving core strength with no impact on the spine is tough, as most core exercises will likely engage the back and spine. I would try to implement standard, good form push ups if you can. Planking is also an option, but again form is king. Do them properly, for less time instead of “cheating” with loose form if you want to make improvements.