603.4. A triggered ability may read "When/Whenever/At [trigger event], if [condition], [effect]." When the trigger event occurs, the ability checks whether the stated condition is true. The ability triggers only if it is; otherwise it does nothing. If the ability triggers, it checks the stated condition again as it resolves. If the condition isn't true at that time, the ability is removed from the stack and does nothing. Note that this mirrors the check for legal targets. This rule is referred to as the "intervening 'if' clause" rule. (The word "if" has only its normal English meaning anywhere else in the text of a card; this rule only applies to an "if" that immediately follows a trigger condition.)
Example: Felidar Sovereign reads, "At the beginning of your upkeep, if you have 40 or more life, you win the game." Its controller's life total is checked as that player's upkeep begins. If that player has 39 or less life, the ability doesn't trigger at all. If that player has 40 or more life, the ability triggers and goes on the stack. As the ability resolves, that player's life total is checked again. If that player has 39 or less life at this time, the ability is removed from the stack and has no effect. If that player has 40 or more life at this time, the ability resolves and that player wins the game.
So to answer your questions:
Does Valakut put another ability on the stack when this happens?
No, only one ability is put on the stack if the "intervening 'if' clause" condition is met.
If so what kind of ability is it? is it a triggered ability that can be copied with Strionic Resonator?
Yes, it is a triggered ability that can be copied.
If it's all the same ability, then does the controller of valakut technically have to choose a target regardless of whether he has 5 other Mountains or not?
Targets are only chosen if the ability actually triggers. (IE. the player has 5 mountains in play.) So, no, one mountain is not enough to kill a phantasmal image.
Note the second part of CR603.4 I highlighted in bold: if when the ability resolves the player no longer has 5 mountains in play, the triggered ability will be removed from the stack and will do nothing. If the ability was duplicated with Strionic Resonator, of course that duplicated ability would also be removed.
Therefore, in the case where 5 mountains were in play and the ability triggers, a targeted phantasmal image would die even if in response their opponent destroyed enough mountains to drop the count and counter the ability. The triggered "this creature dies when targeted" ability would be on the stack at that point and could no longer be stopped if the spell is countered.