so the idea is this, instead of taking an intersection, you take a line.
this gives each individual line 4 liberties, though connecting lines only adds 2 liberties instead of 3.
above i have shown some basic living shapes. good, bad?

You living shapes on the edge are not alive

they aren't unconditionally alive,
but it would be impossible to kill them; barring some weird ko threat battle.

The one on the right is unconditionally alive. The one in the lower left would need the red player to tenuki four nakade moves.

Nope.

This is equivalent to playing on a trihexagonal tiling

Ah, no, you're right. Oops!

I'm not quite sure what you mean by this. If I understand how the topology of the variant board is supposed to work, Red has a five-point eye-space against the edge, which makes this shape equivalent to five on the second line on a normal board.

Hmm, this is tricky, because none of the edge shapes in the example actually close off the edge!