What the mint called an "uncirculated" set from the 1960's, everyone else calls a PL (proof-like) set. True MS coins come from one place only: bank rolls of coins intended for circulation. Everything that came from any type of set is not an MS business strike coin.
Charlton has confused things with the MS-65C and MS-65NC grades, the former being a business strike, the latter being a coin from a set. ICCS has muddied the waters by calling coins MS-65 and MS-65 NBU, again for circulating and non-circulating coins, respectively. The mint itself has confused matters for years by calling sets "uncirculated", which of course they are in absolute terms, but not to be confused with uncirculated coins which originated from mint rolls (business strike coins).
As far as the second part of your question, yes absolutely. If you crack open a roll of coins anything you pull out will be an MS coin. The vast majority of uber-grade modern certified coins on
ebay have been cherry-picked in this exact fashion.
CCCS for example is in alignment with the mint, calling coins from sets "UNC-65" and coins from rolls "MS-65". ICCS as mentioned uses "MS-65 NBU". PCGS commonly uses "PL-65" though generally most coins post-1975 are no longer considered to be "proof-like" in appearance, and are not labeled as such by the Canadian graders (though that can't be said with absolution, some post-1975's have been spotted marked as PL in ICCS holders).
I'd love to say there is a clear and concise answer to the question that you are asking, but the reality is that there isn't. You have to learn the terminologies applied by the different parties and how they apply them to describe different coins, from different periods, with different finishes. There will never be a consensus on the application of these terms, other than to say a business strike coin from a roll is an MS coin.