So, does the Zone Director act as a bridge?
If yes - does it send STP packets?

No, it is just a single logical interface that happens to have 2 physical ports so the connected switches would need to ensure that spanning-tree is observed.

Well if ZD forwards packets from one LAN port to another, it's acting as a bridge.

Let's assume both ports are connected to a switch, then the switch will see its own STP BPDU packets. So, it'll block one of the two ports, avoiding a loop.
Am I right?

Just make sure the switch does that actually and it's not a datasheet error.

It's not forwarding - in a true bridge a broadcast packet coming in on port 1 would egress on port 2. My understanding is that's not what we do - packets coming to the ZD can come in on either port, but they are not re-broadcast. Packets sourced from the ZD exit both ports - so it's more like port replication than anything else.

Purely for simple port redundancy in case of a hardware failure. STP should keep one physical path blocked at all times, but in case of link failure, the other port should unblock and take over. But the STP would all be on the part of the connected switch.