Halting on a (possibly one-way) write-once tape
Ad 1. Write-once TMs are equivalent to ordinary TMs (I think this is an exercise in Sipser, but I don't have the text at hand to confirm) so $\rm woBB$ is uncomputable.

Ad 2. No. The usual zipper encoding also works for write-once TMs, so write-once TMs that never move past the end of the one-way infinite tape are equivalent those with two-way infinite tape. Therefore $\rm owwoBB$ is uncomputable, and the choice of behaviour when moving past the end of the tape doesn't affect this.