And regarding impurity it is always a sin to put yourself at the occasion in this regard.

Well, its degree of sinfulness depends on its proximity and to what it's proximate.  If some occasion is very likely to lead you to venial sin, the deliberately entering into such an occasion would be venial sin.  If some occasion is very likely to lead to mortal sin, then it's mortal sin to enter that occasion.  If some occasion is rather less likely to lead to mortal sin, it could be a venial sin to enter the occasion.  It can also be mitigated by necessity.  So the moral theologians, for instance, will use the scenario of a doctor having to see a female patient nude as necessity requiring entering the occasion, so that entering that occasion wouldn't by itself be sinful ... though any sins that follow would be.  I would simply consult with your confessor regarding any doubts.

If someone were troubled by temptations to impurity, it might be an occasion of sin just for him to go out in public in the warmer summer months when women are often very immodestly dressed ... but you might be required to go out in order to go to work or go shopping for groceries.  But if there's no necessity, such as going to a movie with immodesty in it, depending on how likely it is to cause mortal sin, it could be a mortal sin to go or else venial if it's less likely.  So it depends on the individual and on the circuмstances.

Basically, the principle is that if you will the cause, you will the effect.  If I hold a loaded gun up to someone's head and fire, the person's death is practically inevitable and a moral certainty.  You can't just think, "I mean to pull the trigger but don't will the death of the person."  But if I'm shooting rounds into a crowd for fun, without intending to kill someone, death is not inevitable, but highly likely, so that too would be a grave occasion.  If I shoot rounds into the air, it's less likely, though still possible that someone would die, so there one could make a bit of a separation between shooting rounds into the air (because you want to have some fun) and an accident that could result if it happens to hit someone.  In this case, the death of an innocent person would be a grave evil, so the first two scenarios would be mortal sin in themselves, even if the person doesn't die or suffer grave injury.  In the second scenario, if I shoot into a crowd but no one is injured, I still commit a mortal sin for doing it because it's very likely that someone COULD have been killed or gravely injured  In the third, it would probably be more venial, since the possibility of someone getting hurt would be remote and not reasonably likely.  If the possible effect is less serious, where someone might get slightly injured because of some action I took, even if it were very likely, the action I took would be more venial because the possible outcome would be less serious.

Perhaps a rule of thumb is that if a decision to enter an occasion is effectively the same as a decision to commit a mortal sin.  I know that if I watch some movie that it's very likely I would commit a mortal sin, then watching the movie is the same thing as giving consent to the sin, because you're consenting to the cause of the sin ... even IF, for some reason, you didn't actually commit a sin that time out (perhaps you were interrupted watching it before the "bad parts").  I sit down to watch the film, but then before I commit the sin I'm called away for some emergency.  Just sitting down to watch it would be in and of itself mortal sin, even if it didn't get to that point, or if you got to the "bad parts" and for some reason that time it did not lead to serious sin.

What about going to the beach or the gym? I wouldn't go to either of these because I was curious something.

Though the women dress poorly.