The Wizards I would say yes just because they're the same organization in the same location. Their name change was to promote a more family friendly product rather than one that might be perceived to promote violence.
Not necessarily, because the Bullets won their title while in Baltimore which is not DC, then played in Landover, MD for years after before actually moving into DC. Which leads to another kink in the whole connecting teams to their cities. For instance, the Warriors have been based in both SF (during the 60s) and Oakland (where they won their only West Coast championship in the 70s). As anyone who has been there knows, Oakland and San Francisco are really different cities. Should the Warriors move back into SF as they have explored...Do they lose their title won in Oakland? Most of us would say that's absurd. In this case, I'd argue because they are in the same Census defined Metropolitan Statistical Area or media market (take your pick), we would not really have considered them to have moved. But in the case of the Bullets ne Wizards, it gets murkier. They violate my MSA rule, but not the TV market rule.
Or another hypothetical closer to most Boston sports fans. What if the Pats had or did move to Hartford, CT. Would they be stripped of their history? I mean, Hartford and Foxborough are inarguably different cities. Again, most would likely say no, for the same rationale as the Bullets/Wizards.
My take on this is, give the Lakers franchise their 16 championships. I think its really up to the franchise and the city they are/were based in to decide how to handle the legacy. I think the Seattle claiming the rights to the Sonics franchise history settlement was a great one.