Regarding the initials OP for Dominican laity, it depends on the province.
Here in England, we use nothing. We’re not allowed to use OPL and most of us think that OP is silly, for the same reasons as we’re given by Anita Moore in her blog post. We believe that the Dominican Order adequately understands (although there are always differences of opinion and old fogeys etc

) that all Dominicans are equally Dominican, Lay, friars, sisters, nuns etc. and we appreciate the difference between the Dominican Family and those actually belonging to the Order.
We’re currently looking into legislating on it, in consultation with the Prior Provincial. The main problem is that some Lay Dominicans (one in particular) write to the Press using the initials OP and expressing opinions to which they are entitled but which in no way reflect Lay Dominican policy, and yet in public it looks as though they are speaking for us. In contrast, if a friar or sister were to write to the press, the whole community would be behind them if they were to use the initials OP.
We tend to feel that OP for Lay Dominicans is fine for internal use but that if we are going to be writing in a public forum we shouldn’t use it unless we are actually speaking for the community.
As to when you can use them, anyone who has been admitted into the Order is entitled to use the initials, not just those who are finally professed.