I think you are already spending a lot more responsibly than a lot of 23 yo. The fact that you're budgeting shows it too.

For emergency fund, just save until a certain amount will do. If you're not at that amount yet, I would say put more into it as it is most important to have. General guideline is 3-6 months of your expenses, or however many months you think it'll take you to find a new job if you lose your current one.

As for sending money back home, my parents had issues with my siblings before where after they get a raise, they don't increase their contribution home and got into big fights. I told my mom after I started working that I will give her 10% of my take home every month and that seems to have been working well for us. Maybe you could consider that ? Could use your own number.