I am a random commenter on the internet NOT a doctor.
That said, one of the most important things you can do for a wound like that is not only keep it clean, but also debride it (remove the dead skin/tissue). Don't go crazy and damage the healing areas or remove scab, but that white mushy dead skin is just feeding bacteria. It's gotta go! Scrubbing won't be comfortable but it will reduce the chance of infection and speed healing. Warm water and soap, maybe tweezers. If it starts feeling like the whole toe or part of your foot is red/puffy/swelling/hot/getting worse EVEN A LITTLE BIT then a real infection has set in and you need real antibiotics without delay because these things can spread fast.
As for diving - probably not good for it, probably does increase the chance of weird infection, but I personally would still dive with that injury. Not saying its a good idea, make your own decision. If you do dive, make sure to protect it during the dive and clean it immediately after diving.