What is this “provider” and what role do they play in your system? From basic principles, if you own the gateway, then effectively you yourself are the “provider”
Anyone can create an account with TTN, that is fundamental to it being a public network.
LoRaWAN networks, especially class-A only networks are really optimized for uplink and not downlink. So while you should be able to make downlink work, you really should close the loop of temperature threshold to siren in the node firmware itself, and not via the network. The network could perhaps asynchronously downlink requests to change the configured thresholds though.
Not by itself with the stock firmware, no. There are alternate packet forwarder programs which “sort of” do this but there are many problems in concept, and you’d need to adapt them to the hardware of the gateway. More realistic would be crafting some sort of “splitter” service to run in the cloud and point the gateway at that rather than at either network server directly.
Generally though a give node is only allowed to belong to one LoRaWAN network at a time. If one of the networks never downlinks at all in any way, and based on that you decide that the node should be ABP, then maybe you can have a second network “passively monitoring”. But this would be very unconventional.
Really it seems that the mapping of capabilities offered by LoRaWAN to your goal needs more thought. In particular, try to close the loop between sensor and alarm without going through the network. Or if you must because they are distinct devices, then make the alarm node be a class-C device and chose a network server which supports that (TTN does not).