How do Lasso, The Judge and Belle Star interact with equipment cards?
Lasso and Belle Star would keep permanent (which I think are black-bordered?) equipment cards from having any effect. You can, however, still play one, since those cards don't specifically prevent playing cards. Since removing one by paying cost + 1 gold is an action (according to the Gold Rush rulebook), you should also still be able to do that even in Lasso or Belle Star are active. The cards don't have any effect, but they're still equipment.

I'm less sure about The Judge. It says you can't play cards in front of you or any player, but I think "play" refers to playing from your hand. Getting equipment cards is called "buying", so I'm inclined to say it's a different action and isn't affected by The Judge. I'm not completely sure about that, though, and I haven't found any errata about it.

The rules seem to indicate that equipment cards are identical to other cards in play when it comes to elimination. It does say that all equipment cards are discarded when eliminated, and the rules for Shadow-Gunslinger says "All the rules for the elimination of a player still apply: You discard all your cards in hand and in play (including all equipment)...." That tells me that a Sheriff who eliminates a Deputy would lose his equipment just as though they were regular cards in play.