Yes, Teensy 3.6 works great with the OctoWS2811 board.
The using double insulator pins and sockets are recommended, since Teensy 3.6 is longer. Those taller pins plus the sockets allow Teensy 3.6 position high enough to overhang the RJ45 connectors, as you can see in this photo.
This info & photo can be found on the OctoWS2811 board page (scroll down if using a small screen), with links to the pages for those pins and sockets.
Yes, FastLED can use OctoWS2811 as a "driver", so you use FastLED to do all the color & animation stuff and it uses OctoWS2811 to actually talk to the hardware. You get the best of both worlds... all of FastLED's special RGB rendering features together with the highly optimized non-blocking data transfer of OctoWS2811.
To see how it's done, in Arduino click File > Examples > OctoWS2811 > BasicTest_FastLED.
Or File > Examples > FastLED > Multiple > OctoWS2811Demo.
Or File > Examples > FastLED > Ports > PJRCSpectrumAnalyzer.
Or you can view those examples on github (without having to install Arduino + Teensyduino)
As you can see in the examples, the key to making this work is you first include OctoWS2811.h and define USE_OCTOWS2811, before you include FastLED.h. Then you use FastLED.addLeds<OCTOWS2811> rather than the normal FastLED.addLeds<WS2811, pin, color>.
FastLED can also use WS2812Serial this way.
Do make sure you install the latest version of Teensyduino (1.56). Older versions had some timing bugs with FastLED and Teensy 4.x. This stuff works well in version 1.56.