I got items on WooCommerce and Etsy that I want to get on eBay, will this be able to sync those?
YES, Nembol will be able to move your Etsy listings on eBay, and also your WooCommerce to eBay.
Also, Nembol will sync quantities across the four channels if you set it to do so.
Etsy doesn’t ask for barcodes, period. So I don’t understand how Etsy listings can go to eBay or Amazon
Ebay does not require barcodes, so no problem. Amazon does, so you have to add the barcodes before pushing your products to Amazon.
You may add barcodes in Nembol by duplicating the product and editing it, otherwise you can do it in the source channel (eBay has a barcode field, Shopify and WooCommerce too, Etsy currently does not unfortunately, so you have to do it in Nembol).
Just make sure that your barcodes are GS1 genuine, Amazon checks on that. Otherwise, you may be barcode exempt in Amazon, and in this case you will not obligatorily need barcodes. Check with your Amazon support.