@BillFlowerpot In the first photo you attached, I'm not sure what the problem is. You're correct about Open for Business, and I'm guessing the second answer is supposed to not have quotes. In the second screenshot, that looks like standard captcha. Usually though, if you get one of the photos wrong, it lets you keep trying.
At any rate, you certainly don't need to register to download a mod. Above the description of the mod and below the pictures, you'll see the tabs for Description, Files, Related Uploads, and Comments. The mod files are under Files. There's also a warning to download just one copy.
However, this mod hasn't been updated since 2009, so I'm not sure you want to use it. First of all, it doesn't cover all objects in the game—in the comments, people for example report that the inventing and sculpting stations aren't covered, nor is the fallout from meteors. (Even birthday cakes are apparently not covered.) Also, one person reported that it made the game lag, which would be unsurprising for such an old mod.
There may be a better mod available, or a more efficient approach. If my goal were to end all fires in my active house, I'd buy the Fireproof Homestead lifetime reward, for what I believe is 30,000 points. If my sims didn't have the points, I'd use NRaas MasterController to grant the points, then buy the reward. You wouldn't need an account to download from NRaas either, but then again MasterController has a lot more commands to sort through than a standalone tuning mod like the fire one.
There may be other good approaches as well that I don't know about. Of course, being able to ask about mods means creating an account, either at Mod the Sims or at NRaas. I remember the NRaas site being easy to handle, although there was again one question that I had to answer. You could also ask on the official Sims forums, either in your own thread or in the one for new members. People there will know whether certain mods work with the latest patches and whether they might cause unintended side effects.
https://forums.thesims.com/en_US/discussion/893122/thread-for-new-members-to-post-their-sims-3-game-issues