You can copy any triggered ability, so yes you could copy Hamletback Goliath's and Goblin Rabblemaster's triggered abilities.
You can tell something is a triggered ability based on if it starts with a word indicating that it's stating the triggering condition.
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).]”
So if an ability starts with "when" or "whenever" or "at", it is a triggered ability. This includes Annihilator, because as you can see from the reminder text on most cards with the ability, Annihilator is short for "whenever this creature attacks[...]"
An ability such as the double damage ability on Angrath's Marauders is not a triggered ability; it is a static ability that creates a replacement effect; so you cannot copy it with Strionic Resonator.
Triggered abilities are never on the battlefield. When they trigger, they go on the stack, and when they resolve, they cause some sort of effect to happen. If that effect is something like "target creature gets +1/+1", then often the ability will state how long it lasts, such as "until end of turn".
611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until the end of the game.
This works exactly the same with a copy of a triggered ability created by Strionic Resonator; if the original ability us "until end of turn", then the copy will be as well.
To sum up, the triggered abilities are never on the battlefield and don't last a certain amount of time. However, the effects generated by triggered abilities may have a limit on how long they last; that limit will be stated in the ability itself.
For your specific examples:
If you copy Hamletback Goliath's triggered ability, then there will now be 2 triggered abilities on the stack, both which say “you may put X +1/+1 counters on Hamletback Goliath, where X is that creature's power.” So when each one resolves, you will add X counters, thus giving you a total of 2*X counters. There’s no difference between the counters created by the original ability and the counters created by the copy; nothing will go away at the end of the turn.
You cannot put a counter on a different creature with this, because it is not a targeted ability. It only puts counters on the Hamletback Goliath that has the ability. If it said “put X +1/+1 counters on target creature” instead, then you could choose a different target for the copy, thus adding counters to 2 different creatures.
Goblin Rabblemaster works just the same. You can copy its ability, and if you do, then the Rabblemaster will get more power twice, ending up with a total of +2/+0 for each other attacking goblin. All this power will go away at the end of the turn because the ability (both the original and the copy) says “until end of turn”.
You could copy the ability given by Thornbite Staff, but it would only untap the same creature twice. If a creature has "Whenever a creature dies, untap this creature.", then copying it would mean that the stack now has 2 instances of "untap this creature". So each would resolve, and each would untap that creature. This would only help if you tapped that creature again after the first triggered ability resolved, but before the second did.
You can't untap a different creature because the ability says "this creature", not "target creature". If it said "untap target creature", then you would choose which creature you want to untap when you put the ability on the stack, and you could choose a different target with the copy.
Note that "target' has a specific meaning; it's not simply something that is affected by the ability. It's only a target if the ability uses the word "target"; (or if the ability is a keyword whose rules use the word "target").
For an example where the "You may choose new targets for the copy." part of Strionic Resonator helps, look at Weaver of Lightning's "Whenever you cast an instant or sorcery spell, Weaver of Lightning deals 1 damage to target creature an opponent controls." If you cast an instant while controlling Weaver of Lightning, then you could use Strionic Resonator to copy the triggered ability, and pick a new target for the copy so that you deal 1 damage each to 2 different creatures.