Your intuition is correct. The Dash action simply increases your movement, but doesn't actually increase your Speed.
If your speed were 30 ft. and you decided to Dash, you'd still only spend 15 ft. (of the gross 60 ft. of movement) to stand up from Prone.
If you had an item, effect, or class feature that increased your Speed (e.g. the Boots of Speed, which double your walking speed while active, or the Mobile feat), then standing from Prone would cost more, according to the rules as written.
The invisible or hidden condition ends when the attack hits or misses.
From PHB p. 194 - 195 regarding Unseen Attackers and Targets
If you are hidden -- both unseen and unheard -- when you make an
attack, you give away your location when the attack hits or misses.
That means that your second interpretation fits better.
Using only their turn would suggest they are able to stay hidden until
they spring to make their attack.
That takes the answer to your question away from an abstraction, and ties it to initiative/turn order. (D&D 5e is a turn based game, in its implementation of combat). It also preserves the advantage that accrues to such an attack.
the creature's attack rolls have advantage (p. 291)
Comparing the text of the invisible condition (p. 291), and the invisibility spell(p. 254), is consistent with the cite on unseen attackers and targets.
The spell ends for a target that attacks or casts a spell.
Subordinate question: will movement give the invisible/hidden PC away?
Maybe. Unless the PC moves, or declares a move, that question doesn't arise. (Unless you want to go out of your way to complicate things. Each table has their own preferences on that).
The chance of discovery depends on how soon in the initiative order that PC gets their turn, and what any other hostile creature does or doesn't accomplish via a passive or active perception check. It will be dependent on the situation.
For example, a creature like an Ancient Silver dragon, with a very high passive perception score, may detect the PC such that now that PC is no longer hidden from the silver dragon. That can render the entire effort moot, but regardless of whether or not the creature stayed hidden, the attack if the PC is invisible is (a) made with advantage (per the condition rules, p. 291) and (b) ends the invisible condition for the PC.
Best Answer
A hidden character may move at full speed in combat.
The idea of “slower speed stealth” comes from the rules for “Travel Pace”:
However, we see in the Sage Advice Compendium:
So the travel rules that mention moving slower during stealth do not apply, and since there is no rule for combat about stealth hindering movement speed, your movement speed is not hindered while stealthing during combat.