Sorry if I repeat a bunch of stuff already mentioned. I haven't caught up on the whole thread but these are just my off-the-cuff thoughts!
What are some tips and tricks you might have for balancing care of the child/children, your livestock/pets, and household chores.
Baby-wearing seems to work for a lot of people, and my sister basically papoosed her now 5 year old to handle 4 other kids and various pets plus chickens. Even if it is just to get your hands free for making a sandwich, that can be really nice. It super helps if someone that has done baby wearing can help you and baby into the getup and make sure you are both comfy and safe while you learn.
My sister and I were both fans of baby-fencing. I only ever used mine in the house to create safe play zones/keep the dog out of baby-ville but she would take hers outside in a shady spot if I recall correctly with a sheet on the ground. This enables light gardening or a few nearby chicken chores. You can buy multiple sets for much more space than a traditional play pen. I even used them for improvised chicken isolation pens later on when I no longer needed them for kids.
Baby gates for everything, especially the dog if you have a bigger one

Total life saver not worrying about unsupervised activity between baby and dog if you have one as part of the household.
I also had a monitor I could wander a bit with for quick outdoor trips during crib naps. I could turn it up loud and still get reception on my porch and stay nearby for short chore sessions (but I keep my chickens near the house so your mileage may vary with this one).
This is more general advice, but everyone does stuff differently! And every baby is so unique imo. I had two that were night and day, so it probably isn't always that extreme, but it is OK if people are like, "you should do this" but you find it doesn't work with your individual case.

You are doing great anyway, just keep trying things until you do find what suits your situation!