Most missions have no LSC access anymore, which is why when people pine for the Dubsta 2, it is done within a mission, and you must drive it to your garage, instead of tracking and insuring it right there at the LSC where you found it. The Lamar missions (or at least Funeral Party) LSC is accessible. I've noticed, due to recent outfit glitches....the accessories selection in mission are disabled as well. It may well be that I was playing them while VIP, or had a VIP outfit selected before entering....but I believe that is new as well. I know I was VIP all day yesterday, but in mission I couldn't switch any items via menu, nor equip armor.
Your properties never appear on the map during missions, but when you arrive at them the blue entry circles appear. Maybe that's what CCali
was thinkin of. As far as I know, LSC has been locked in missions for some time, but in other game modes, the door will open, but no interaction is available.