You can't take both Dash and Ready actions
Normally you don't "prepare a reaction". You take the Ready action instead. See Player's Handbook, page 193, "Actions in Combat"
Sometimes you want to get the jump on a foe or wait for
a particular circumstance before you act. To do so, you
can take the Ready action on your turn so that you can
act later in the round using your reaction.
Then you use you reaction (if still available) to actually perform the readied action when trigger occurs. You spend both your Action and your Reaction in this case. Therefore, you can't use your action to Dash after you declared a readied action, because it is already spent.
Unless you have an extra Action, or you're a Rogue
As notices by @NeilSlater, a Rogue has his Cunning Action feature, which allows him to use a bonus action for Dash. You can take the Ready action afterwards. However, this doesn't give you more movement than usual, since you just can Dash twice.
Also, you can't use "my turn ends" trigger
PH requires "perceivable circumstance" for a trigger:
First, you decide what perceivable circumstance will trigger your reaction.
Turns are game mechanics, they don't exist in the game world, hence, are not perceivable by the character, see Can you ready an action for "immediately before my next turn"?
You can move up to your speed using the Ready action
However, the question remains — what if you spent all your movement, and then take the Ready action to move 30ft more? Yes you can do this, providing your speed is 30 (or more) feet:
you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it
Ready action works like a "delayed Dash" in this case.
Best Answer
No the 120' per 10 minute movements doesn't include searching for either traps or secret doors. On page B21 of Moldavy's Basic Rules for D&D and page B22 both actions (secret doors, traps) require search of a specific area (specified as a 10' by 10' on B19) and the search takes a turn (10 minutes) to perform.
The searching referred to in the movement rules probably refers to non-obvious details like a grate at the bottom of a wall, bones littering the floor, etc. I got this impression from the reference to a 'dark and dingy' dungeon on B19. And from reading the Dungeon Master information on stocking the dungeon starting on page B51