I'm sure that harry remembers that the horcruxes are difficult to destroy. He may not mention in specifically.
He tried to destroy the diary by burning it in the Gryffindor common room fire, though, which doesn't seem like a plausible mistake for someone experienced with Horcruxes' durability.
Regarding Harry arranging them to be destroyed immediately, he is trying to keep his future knowledge a secret and going around destroying things he has no business knowing about yet would be a bit difficult. Not to mention, even if Voldemort doesn't know when a Horcrux is destroyed, he may have protections on the place it's currently housed or periodically check up on it.
In that case, Harry shouldn't have had Sirius remove the ring and the cup from their hiding-places, but he apparently didn't expect that to be a problem. (Actually, he did expect it to be a problem with the Cup specifically, based on past/future experience, but Sirius was clever enough to get around the alarm Voldemort had placed on it.) The locket, of course, had
already been moved from the sea cave where Voldemort hid it to Number 12 Grimmauld Place by Regulus back before Voldemort's mishap at Godric's Hollow, and Voldemort apparently never noticed.
Yes, the horcruxes are dangerous, but only in the presence of magical people interested in interacting with them. If they are not currently harming anyone in their current location, why not leave them there?
The present location of three of them is Arthur Weasley's shed behind The Burrow. Granted, the Burrow is nearly impregnable now, but "nearly" isn't the same as "perfectly," and the Horcruxes make it even more of a target whether Harry is home or not. They're unlikely to be able to ensnare Sirius or any of the Weasleys, but given their function and the ultimate reasons why Harry needs to destroy them all, I don't really see a persuasive argument for waiting until
after Voldemort rises again to dispose of the things.
Also, perhaps Harry doesn't want to subject someone to the mental anguish that is sometimes involved with destroying them until he is stronger magically and wiser/older.
Now that
could be a good reason, although I doubt Sirius would find it convincing (he survived twelve years of the mental anguish that comes with unshielded exposure to Dementors, after all). I think it would be worth mentioning it in the text at some point if that's why they're waiting, though.
Exploring that the pureblood faction is more dangerous without Voldemort around could be done after he is destroyed. It's an interesting topic, but some of the story themes include "things Harry tries to change that still don't change". Perhaps Voldemort's return to the flesh is one of those things
I suspect it is, but I'm unconvinced it should be. Ah well; it's not like I'm going to stop reading, and loving, every new chapter the Authorlord puts out -- however long it takes (if I can wait on George R. R. Martin, I can wait on Viridian as well).