I think 1 or 4 would be the best solutions. Honestly I don't think speedruns.com hosting score attacks would really confuse too many people.

Also if you did just add it to the existing site you could potentially have it so that being on the high score or speedrun pages for a specific game would have a direct link to the opposite page. So for example if I were looking up Contra speedruns I could also just click an option on the sidebar and go to the matching "score attack" page for it, if it exists.

If you really want a new category so bad the only thing that seems to make sense to me is 80 Socks, maybe it'd be an extension of cheat% since otherwise it'd just be a pseudo Any% run that stops collecting when it can get to the Flying Dutchman's level.

My compy isn't stronk enough to stream without minutes of stream delay. If I ever do stream it'd most likely solely be to record a speedrun attempt whenever I actually get around to it.

I don't remember if I checked the shop after the Drill Missile but since that stage actually uses the silver blocks as a puzzle element (for those bombs hidden in blocks) the earliest the game would give it to you is right before the final stage. I'd have to play through again to find out but FAQs i've looked at seem to say it's only after Croog, but you can get the purple drill bit to go through the curved grey metal blocks before then.

Having a NG+ 100% category makes sense for the normal run and we could argue about it for Treasures but the overall length seems like it might dissuade people from running it and I think it'd make it tiring to watch. Someone more well versed on the treasure locations can correct me if i'm wrong but I believe you can only find 3 or 4 of them without replaying stages, thanks to the first upgrade only becoming available after most of the treasures that require it have been passed in the story progression. I agree with you on the health upgrades though and since there's no way to waste chips (unless i'm forgetting something?) starting on a fresh file is the only realistic way to keep that aspect intact. There's probably a way to just cheat chips in but I don't know how and why bother if that's not the route being taken anyway?

If memory serves, the lion's share of treasures are locked off to you until post-game because they're behind the harder silver blocks you need the hardest drill bit to get through and that drill bit doesn't unlock until you beat Croog. Since you essentially need to play through the game twice with the current ruleset, consider this:

Start on a save with the game completed and all extra maps/drill bits bought, you can open your treasure menu to show you only have the Red Diamond in your treasure list, time starts when you select a stage and confirm, time ends when you finish the stage of the last treasure you needed.

While this does remove the chip collection aspect of the first run, it might make it less daunting to anyone looking to run this category. As soon as I hunt down the right version of VBA allowed for speedruns I could probably do a playthrough to make a save for this if there's interest.

Edit: Also, should I just get all of the health upgrades or leave it at default? I would assume i'd get them all but I don't know.