Current through an inductor is like a train. It has inertia. If you try and stop it, it uses its stored energy to try to not stop. For a train, this is the energy stored the velocity of its mass (inertia). For an inductor, this is the energy in the magnetic field.
You can stop a train very quickly by placing a wall in front it and the train will stop but not for lack of trying. It will attempt to smash through the wall and if the wall is not strong enough it will cause a lot of damage. Or you can stop the train gently over a longer period of time and not incur so much damage.
When you try to interrupt or reduce the the current in an inductor, it expends the energy stored in its magnetic field to try and keep the current at the same level. If it takes a lot of "push" to shove the current through whatever new obstacle you presented (like an open switch) then the energy is used to generate a very high voltage to push that same current through that obstacle. That voltage can cause damage. In other words, an arc is is generated to jump across or blast through the switch you are using.
But if you give that current an easier path to follow so it loops back to the inductor, then it does not need to generate such a high voltage to keep that current circulating within itself so that the current dies down more slowly and gently.
Question 1: The motor is reversing and it requires two Schottky diodes around the motor (since polarity changes). Can I just put one diode right outside the PWM controller before the reversing relay where polarity doesn't change?
You worded it a bit strangely but the concept is sound but the implementation is not. You put the flyback diodes in parallel with the switches in the direction that DOES NOT cause a short-circuit given the power supply voltage, and you need four of them. Together, they provide an easy path from one motor terminal to the other for flyback currents to flow so that a flyback voltage spike is not produced, while not actually shorting the motor out in any particular direction thereby allowing bidirectional drive. When spinning in one direction, a diagonal pair conducts the flyback current. When spinning on the opposite direction, the other diagonal pair conducts the flyback current.
simulate this circuit – Schematic created using CircuitLab
Question 2: In either scenario, can you recommend specific diode to buy? it's 24v 60amp (max for both controller and the motor).
Shopping questions are off-topic here but at 60A...maybe go big. But relays are fairly hardy so doesn't need to be that big.
Question 3: the reversing relay activating coil is 12v and its turned on by another relay that is linked to Arduino (operating at 5v). When turning off the bigger relay, i get lots of "feedback" that messes up the rest of the 12v and 5v electronics (DACs, sensors, etc.). So, was also thinking about putting a diode across the reversing (larger) relay coil. Does it make sense? If yes, what is the specific diode I should buy?
Yes. This provides the flyback currents an easy path to flow around the inductor so that it does not generate a huge flyback voltage spike when current is interrupted. The reason you can do this here but not the motor is obvious: the current here is only ever flowing in one direction through the inductance so you know which direction you can place the diode and not have it muck things up. But current can be driven through the motor both ways since you want bidirectional rotation and placing diodes directly in parallel with it would short the motor out in one or both directions.
Schottky diodes are good for this for a number of reasons if you can find one of the appropriate voltage and current ratings. Note: You don't need a 60A diode here since it's not actually conducting 60A all the time.
That said, you have numerous other problems with your design.
Your MCU pins probably cannot supply enough current to drive even the smallest relay. So your "Activate Reversing" relay probably won't work. Probably need to drive the relay with a transistor and drive the transistor with the MCU. This relay coils also needs a flyback diode since your MCU pins (and the transistor you use to drive it) are particularly vulnerable. Far more so than the relay you were asking about.
Simple transformers convert AC-AC, not DC-DC so you can't power your Arduino the way you have drawn.