Are there any well-engineered, non-synthetic, t-shirts?

Over the years I've accumulated a messy collection of various old cotton t-shirts. Each with its own quirks. Some small, some uncomfortable, some not breathable, some with a weird smell after lying not-used for too long in the closet.

I just want to buy 10-15 identical t-shirts, and not have to think which one to wear for what occasion.

Are there any t-shirts which fulfill all, or most, of the following requirements?

- comfortable

- long-lasting (does not stretch/shrink too much over time)

- not staining other clothes during washing (so, probably some neutral color)

- breathable

- easy to wash and dry (30-40*C machine-washable, ok for tumble-drying)

- non-synthetic (e.g. cotton, linen, wool)

Purpose:

- wearing at home through the day as base layer

- running/gym

- sleeping (as pyjamas)

Color/brand doesn't matter, but eco-friendly manufacturing process as whole would be a good bonus.