1. Any nRF52 based device should be able to run ANT as long as it has a 32.768 kHz crystal with +/- 50PPM or better accuracy. If you're planning to interface it with Arduino as an external network processor, you will need your own UART drivers to interface using the Network Protocol Serial Interface spec. I've seen one project pop up on the forum recently (
https://www.thisisant.com/forum/viewthread/6531/) but I haven't had a look at it myself yet.
2. Technically you could use the nRF52 as a standalone SoC without the Arduino, but it would require familiarity with C and embedded programming. This also depends on how much data you need to store. Each sensor transmits 8 bytes about 4 times a second, storing this data stream raw might not be a good idea depending on how many sensors you're recording from and for how long. The nRF52832 has 512 kB of flash, but around 1/3 of that will be taken up by your application code and the S212 SoftDevice. Arduino's come in many sizes.
3. If this device cannot be Android or PC based, then you should get familiar with embedded development and some ANT/ANT+ basics. SimulANT+ and ANTware are PC tools meant to assist with the development of the ANT/ANT+ wireless messages. SimulANT+ can accurately emulate ANT+ sensors/displays, while ANTware is used to play with generic ANT channels.
If you could use the PC/Android tools, it could be as simple as plugging a USB stick into the device and using SimulANT+ or ANT+ Plugins on Android to decode the sensor data for you. Typically this is where most developers start or do their sanity checks on, but you'd need an ANT USB stick.
Also, it doesn't look like that breakout board has a JTAG programmer. You'll need to already have one yourself. If you didn't mind the cost, the D52DK2 has all of the parts you'd need to get started.