Timing/Stack question about abilities triggered during combat
Yes, both are triggered abilities and can be responded to, e.g. countering them with Stifle. If the ability triggered, it's too late to counter the ability by destroying the source with e.g. Naturalize. Even though the artifact is gone before the ability resolves, the ability itself is now a different 'object' and won't be affected by this.

603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”
603.2b When a phase or step begins, all abilities that trigger “at the beginning of” that phase or step trigger.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.