hip flexor stectch? would this help hamstrings as well?

What are some of the most effective ways / exercises you've found to work on your deep core muscles? Are there any you can do while in your chair? Please share, thank you

what are some you do from lying or standing?

I like the plank ,and crunches. I see you have a pull-up bar. Another good one is to hang from the bar and pull your knees up towards you chest. These are pretty tough.
Once you find your core keep it tight and it will help with many other exercises.

That will work just as well. Start easy even if one rep, rest do another, I do most exercises in reps of three. It's important to rest between.
I just want to say I'm just explaining what I've been shown and what I've learned over the last five years. Other members have suggested the site physiotherapy something or other and I checked it out and it's a great site.
You can put in your injury level and it will give you exercises for that level of injury. The most important thing is attitude, you have that going for you so all I can say is drive-on.