2to be sure, talk to your advisor first.
the main concerns are 1) have you used ideas that are originated from the
lab/advisor (even remotely)? 2) are there any other postdoc/student working
along the same line that you may potentially scoop their papers? and 3) are
you 100% sure you are using your own resources/personal time for this work?
if you are not sure about any of the above, it warrants a discussion
on the other hand, a discussion does not mean you won't be able to publish -
your advisor may well just tell you go ahead and there is no issue.
4如果是fellowship， training grant，感觉管理比较松。如果是项目或者老板自己出