As long as he was validly ordained..yes.

Traditional bishops were not consecrated, or chosen to be consecrated, by anyone with authority. They are simply preserving the priesthood and apostolic succession. If a priest has disagreements with a bishop they are associated with I think they are free to go "independent", the Bishop does not have any authority over them as far as I know

 

Trad bishops have no authority, no jurisdiction even if elected or selected as superior of such-and-such society or congregation because trad religious congregations have no canonical personhood or standing contrary to any trad bishop's claim. Trad bishops, all trad clergy are merely sacramental pez dispensers during this time of crisis in the Church.

And, yes, the priest's sacraments would be valid. In fact, invalidating a sacrament takes a little effort since one would need to recompose the sacramental form or seek out matter other than what is prescribed.

Yes, if the priest/bishop is valid, so are his Sacraments. I could be wrong, but I do not see how a priest/bishop without “normal” jurisdiction has the authority to bind the consciences of laity to only receive Sacraments or hear Mass from himself and/or his priests.  If said bishop is running a seminary and ordaining priests, I do believe he may, by authority of natural law, require his priests/religious to conform to his or his organization’s position on different matters. Of course, if a bishop insists upon binding the consciences of lay people, then he can expect to serve a very select, limited group of people, the more restrictive the rules, the smaller the group of parishioners. At a certain point, he will end up with a cult of personality that will splinter in many directions and/or die out with him. 
Conversely, no rules or total laxity will result in anything goes, no difference from the world, in which case, why bother at all?