The core concept of GVT is to progress in volume and eventually hit 10x10. You'll have a better idea of where you're starting from by the end of your first week, but I'd stay open to possibly needing to run the program longer than 4 weeks to get the full effect. Like, if after 4 weeks you're not yet hitting 10 reps on the last 3 sets of an exercise but you ditch the plan anyway, that's progress left on the table.
Any kind of easy-ish active recovery should be fine. It's hard to walk too much, especially if you're currently carrying visible body fat. Anything higher intensity, like HIIT, gets tricky because you don't want to turn it into a hard conditioning session.
Eat much. Don't avoid carbs or healthy fats (or, obviously, animal protein). This would be a solid guideline.
I'd suggest plain old boring GVT as your intro to the program. Same split you wrote, but only one exercise per bodypart for 10x10. Done. At most, you could add one accessory exercise per body part for 3x12. Get more out of less. The more simple, the better.