If you have allergies, I was told it is best to avoid all carpets.
There is nothing wrong with used carpets as long as they have been cleaned.
Two things you need to consider when buying carpets are what kind of flooring you have and do you have underfloor heating? If the back of the carpet is not antislip, you need to buy this also.
Some carpets cannot be cleaned, not even with water, so best to avoid these.
It's not really a case of avoiding carpet to avoid allergies, but if your child has a specific allergy it could be to wool, dust mites, cleaning chemicals, chemicals in synthetics....
For removable, washable carpets I would go with cotton but it might be treated with preservatives when you first get it, it will pick up dirty marks quickly, need regular washing and not be very sound proof. Wool is great for durability, cleans pretty well but only with a carpet cleaning product or small amounts of warm soapy water for spot cleaning,and feels nice to touch usually. Per have real woolen carpets in our chilcare where i work, and occasionally we have a baby whose skin reacts to the wool for those babies, we don't put them on the woolen carpet, and we have alternative cotton mats.
Synthetics are pretty durable and may or may not stain depending on the specific carpet.
Personally, I don't think it really matters. Your budget and personal taste will matter. When the kids were little we purchased cheap mats and rugs because we knew they would get very dirty...and plan to upgrade as the kids get older. We do not have under floor heating and our flooring is mostly either high traffic commercial carpet in the central area, and wood laminate in the rooms, which is pretty water tolerant. With carpets you do have to watch out that spills are not left to soak through and wet the floor underneath...