1. Does PADI allow their instructors to use the standard DIR gear while conducting PADI OW and Specialties courses?
Yes, DIR/Hogarthian type equipment fulfills the standard for instructor equipment according to PADI.
The specific standard for instructor required equipment is containing in the PADI Instructor Manual, General Standards and Procedures chapter:
"Equipment
Standard Diver Equipment
Make sure divers have, at a minimum:
1. Fins, mask and snorkel
2. Compressed gas cylinder and valve*
3. Buoyancy control device (BCD) with tank mount or separate backpack, and low pressure inflator*
4. Primary regulator and alternate air source*
5. Breathing gas monitoring device (e.g. submersible pressure gauge)
6. Depth monitoring device
7. Quick release weight system and weights (if necessary for neutral buoyancy, or if required for skills practice)
8. Adequate exposure protection appropriate for local dive conditions.
9. At least one audible emergency surface signaling device (whistle, air horn, etc.).
10. Dive computer or RDP (eRDPML or Table)
Standard Instructor and Certified Assistant Equipment
Have all standard diver equipment plus:
1. Time monitoring device
2. Compass
3. Knife/diver’s tool
Exception: Where prohibited locally.
4. Two surface signaling devices – one audible (whistle, air horn, etc.) and one visible (inflatable surface tube, flare, signal mirror, etc.).
5. Dive flag – where required locally
6. Instructions for use for dive computer or RDP/eRDPML"
None of these standards/descriptions preclude DIR/Hogarthian type equipment. A BP&W remains a BCD as per the function and description. There are no prohibitions on long hose. Indeed, the skill of air donation requires the AAS to be available "in a triangle between the mouth and hips". From the mouth donation is also acceptable and standard for divers using AIR2 type devices.
2. Do they still require snorkel?
They have to "have" a snorkel. i.e. be equipped with it. They do not have to wear it on their mask, other than to specifically demonstrate certain snorkel-focused skills. For that reason, some instructors opt for folding-type snorkels that they store in a pocket etc.
3. Can PADI OW students use a BP/W and Long hose (DIR SETUP) while learning OW?
As per the equipment standards quoted above.... Yes, they can wear DIR/Hogarthian type equipment.
I taught OW courses for a year at a dive center that only had BP&W BCDs for rental and classes. I wear a BP&W with long-hose regulator (DIR) for any course dives where I'm not using sidemount and/or doubles - but I don't really teach many/any OW classes any more.