Is coop required for progression?

Co-op is not necessarily required but it softlocks your ability to gain 5* weapons as a F2P. Each week you can get up to 150 5* Weapon Shard (you need 450 Shards for a weapon). If lag is your problem, join the 'stable' level coop and let the team carry you. Just deal a bit of damage so the players don't think you're AFKing.