There is a minimum, not at protocol level but at the node level which is set by default to 0.00005. Any fee under that is not relayed by the node.
Also, the stress tests do not send transactions with 0 fees. They always send with enough fees.
Well i`m sure one of the previosu stress test were done with 0 fees and have flooded the mempool almost making the bitcoin network crash.
It was a quick fix of it, but it was concerning since the transaction didnt even took place, it was the miners memory that was targeted.
It was more like a DDOS against bitcoin.