Sequential shifter & pretty sure its 5 speed. No Idea on brand, could be anything, Cant hear any trans whine to indicate possible straight cut gears. Nice motor, got nothing below 4000 RPM though where it fell off the cam around 11 minutes. If that's open track day laps at Le Mans you can have it, I would feel safer with actual race conditions, would have a better idea of what the other fellows about to do next in a race. Some pics on the facebook link and second video suggest car is space frame and transaxle 'looks' a bit like one of the 'ELITE' versions Ive seen pics of IIRC.
If anyone here knows how to 'lift' a frame or two off the video there are a couple of good overhead shot of the transaxle just after start of second video. They should also reduce some concerns some folk have of CV angles.!!