10x3 for Fat Loss might be a little intense if you're not used to consistent training. Plus, you'd want to find an alternative for all that jump rope if you've been having foot issues. I'd get with something more basic, along the lines of the "Sample Conditioning Workout" in this article, 3 or 4 days per week.
That would be a great intro/reintroduction to hard training while covering all bases with an emphasis on fat burning/"conditioning".
What exactly does an "underdeveloped core" mean? Do you have chronically weak postural muscles? Are your abs/low back atrophied and smaller than they should be for someone your size? It's hard to work around or correct an issue if it's not clear what we're dealing with.
That said, if you need direct core work, there are plenty of core training plans on the site that can be incorporated into most programs. The most recent was just a week or two ago:
Is that on purpose (as in, you tell yourself not to eat much/often) or just incidental (like you're just not hungry or used to eating more often)?
Follow this step-by-step outline to get your diet in order, without stressing over too many things right off the bat.
Definitely. It'll be a key part in helping you to retain/build lean muscle. Something like Surge Recovery is a basic post-workout shake. Don't freak that it's relatively-higher in carbs. It's supposed to be, and it's been used by tons of people during successful fat loss programs.
As was said, that's normal, especially when you begin a new weight training routine. Soon enough your body will adapt, but a post-workout shake will definitely help you start recovering sooner.