Can I play this Silverbolt with the old Silverbolt on my starting team? His start of the game ability says I can’t put the original Silverbolt into my KO, but it doesn’t say anything about his teammates.

Yes, you can, but you will not be able to combine if you have both because they represent the same piece for the Combiner. For example, you could have a 12/8/5 team consisting of both.

Does the bot mode ability on Silverbolt still trigger while my opponent has an Autobot Jazz and Here’s the Thing in play?

No, all Black pips would be considered as White and the Silverbolt, Air Commander ability would not be able to trigger due to lacking the minimum flipped icons requirement.