"1. What kind of clear silicone can I use to fill the gaps?"

Any 100% clear silicone will work, but this should be done by the installers, not you. You don't want to give them the opportunity to say you screwed it up.

"2. How long does it last before I have to scrape it off and re-apply?"

The silicone that's doing the gasketing job it's meant to do should have been applied to the sink flange liberally enough that when the sink was clamped tightly to the bottom of the top, it squeezed out. The squeeze-out should be tooled. Any additional silicone as you are applying is topical and cosmetic in nature. Peek under your sink. If you see no silicone squeezing from the flange, have them remount your sink properly.

Your top looks like estone, but in natural stone tops that have been rodded, improperly gasketed sinks can cause catastrophic rod failures which I charge $800=$1,600.00 to fix and you can usually still see the crack.

"3. Is there a clear silicone with antibacterial features? The sink will be exposed to a lot of germs?"

Keep it clean and it will be fine.