Check the label to be sure, but I'd say mostly yes on washability. Just be sure to wash and machine dry your swatch , or sleeve, before
taking gauge for the sweater body. Cotton has NO bounce-back and can actually grow after laundering because of this.
I stay away from larger garments in cotton, the weight of the garment wants to pull and stretch out the knitting and it doesn't spring back, and it can be a frustrating fiber. But for a baby sweater scale it can be great.
I'm partial to eco-fibers, so that's why I list organic cotton first. I think linen or hemp would probably beat cotton for eco footprint, and also for machine wash durability, but they won't be baby-soft until they've been washed and washed and washed. I've never knit with tencel, but that is the most eco-friendly of the man-made cellulose-based fibers, if you're leaning in that direction. Mercerized cotton is soft, but I believe uses a lot of chemicals to get that way, so I'd avoid that. Bamboo and soy fibers can be quite soft, but again use a lot of chemicals as I understand it. Knitting Green
by Ann Budd is a great book on these topics, for the curious.