"Kai" can mean: "and", "also", and "but", too?
Yes, all of that, but usually a simple "and" will do. Here, the translation "while" or "meanwhile" probably wouldn't be far off.
The first sentence is usually taken as an imperative, e.g., "But go ahead and strive for the greater gifts" — which seems to make sense in the context. In the second sentence, since it obviously refers to the upcoming chapter, you can take δείκνυμι as an immediate future: "I'll show you," "I'm going to show you," etc.