What kind of caulk should I use around the bathtub

Typically no caulking is used when setting a range top into a counter, however in your case where you are getting water from the sink under the lip, a small amount of silicone would not be a problem. The very outside mounting flange of you cook top should not get very hot. If you can touch it when the range is in use, then a standard silicone would be fine. If for some reason you determine a high heat material is needed, a fire rated caulk is available right off the shelf at most home improvement stores. It is usually red in color and used for sealing holes in walls and floors where a fire rating is required. Use a very thin bead of caulk and wipe off excess immediately with a damp cloth. One caution, silicone can stain some types of natural stone counters, so if you have marble or some other type of stone, test the sealant in an inconspicuous place before using.

Yep, use a paintable latex bathroom caulk. We always use PolySeamSeal (now owned by LockTite, available at Lowe's).