Do I need a 3 pin to 4 pin adapter for the motherboard fan connectors

Actually, the 3-pin connector is newer.

1. The 3-pin connector provides power to the fan, and optionally (if the fan supports it), monitoring of the speed of the fan.
2. Does your fan have two wires/connectors‽
3. If the fan supports monitoring, then you get the added benifit.
4. Not as such. There may exist special adapters, but it’s better to make use of separate connectors (most boards have three). If the fans support speed-monitoring, then you definitely can’t daisy-chain them (or at least not expect it to work, and possibly damage something).
5. Nothing. Speed-monitoring and speed-control are separate. The third wire in the fan allows it to “report” its speed to the motherboard. Obviously connecting it to the two wires of a four-wire Molex connector won’t support that. The motherboard controls the speed of the fan by adjusting the voltage it provides to the fan. It can only do that if the fan is directly connected to the board, not if it is connected to the PSU.
6. Fan speed-monitoring and control allow you to automatically raise or lower the speed of the fans as necessary, either via the BIOS, or a program like SpeedFan. This allows you to keep them running fast for more cooling when the system is hot, and keep them running slow (or off altogether) when the system is cool; providing you with a quieter system (fans can be quite noisy).

Well, all three options are possible.

The chassis fan option is definitely the best option - it will do speed control, and will keep the fans running at the required speed. Where possible, this is the best option. Use the chassis fan controllers first

The molex/fan connector connected to a molex connector will have the fan running at full speed all the time, unless you have a fan controller. You'll notice the 'converter' you have only has two pins, since computer fans gracefully degrade down with lesser wires - 2 wires are fixed speed, 3 wires have PWM for speed control and 4 wires also do speed feedback. This will as such, run your speed controllable fan at a single speed, controllable by voltage

As for pwr_fan - various sources source 1 source 2 say that it won't control the fan speed and Its proper use is for monitoring PSU fans where supported. If you have more fans than chassis fan connectors, you could presumably use this as it should be possible to power a fan off it, however. Practically this would be preferable if you didn't want to use a molex connection specifically for fan power, but its not a replacement for a proper chassis fan supply.