Is it normal to be healthy without protein?

I doubt you only eat food with protein content only twice a month. Most food has a least some protein in it. An average serving of broccoli, for instance, has 7 g of protein. I think you are getting more than you think!

Actually, we get protein from veggies too, so you may get more than you think. It IS a myth that we need meat to survive.

It's a myth we need as much protein as they say we do, in most cases. As with every other nutrient. Some of us have ample B12, some have none. It varies due to age, body type, weight, pedigree, and lifestyle.
You seen to need very little, but id either confirm with a doctor (as lacking a needed nutrient can be harmful), or add in a bit of protein food/powder at least once a week.

You need to take a course in nutrition. This will catch up with you and trust me when I say it will hurt.

No, you do need it for your muscles.