For your first question (what happens if Brink of Disaster is played on a tapped permanent) - nothing happens to the permanent enchanted by Brink of Disaster until it goes from untapped to tapped (via activating an ability, attacking, or something like Twitch). The game event of it becoming tapped is what triggers the ability.
603.2. Whenever a game event or game state matches a triggered ability's trigger event, that ability automatically triggers. The ability doesn't do anything at this point.
As for your second (what happens if a permanent's activated ability cost requires tapping, and you use it), events occur in the following order:
- The activated ability is put on the stack.
- The active player would receive priority - before this, the triggered effect is put on the stack on top of the activated ability.
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 116, "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.
- People respond normally (active player now has priority, can pass, lets things on the stack resolve normally).
- Unless something else is put on the stack causing Brink of Disaster, it resolves, destroying the permanent (unless it's indestructible).
- People respond normally.
- Unless the original activated ability was somehow removed from the stack, it resolve.
In the case of step 6, the activated ability will still resolve, putting a +1/+1 counter on each artifact creature you control.