Freedom of Movement states:
The target can also spend 5 feet of movement to automatically escape from nonmagical restraints, such as manacles or a creature that has it grappled
The Rogue Scout can, through their Skirmisher Reaction:
You can move up to half your speed as a reaction when an enemy ends its turn within 5 feet of you. This movement doesn't provoke opportunity attacks.
Can part of your Skirmish half your speed be the 5' from Freedom of Movement?
Just to clear things up for the rest of this answer, Jeremy Crawford has pithily stated the difference between the two in this tweet:
Speed ≠ movement. Speed is a potential. Movement is an actuality.
Let us look at how using freedom of movement works on your turn normally.
The PHB states a rule that allows you to, on your turn, move up to your speed:
On your turn you can move a distance up to your speed...Your movement can include [...]
Thus, a rule has granted the character the ability to move on their turn only.
And, then you could obviously use 5 feet of that movement to use this part of freedom of movement if you so desired on your turn:
The target can also spend 5 feet of movement to automatically escape from nonmagical restraints, such as manacles or a creature that has it grappled
We established above that the general rules establish that you can only move on your turn. However, there are features that override that general rule such as Skirmisher Reaction:
You can move up to half your speed as a reaction when an enemy ends its turn within 5 feet of you. This movement doesn't provoke opportunity attacks.
Note the similarities between this specific rule:
You can move up to half your speed as a reaction...
and the general one:
On your turn you can move a distance up to your speed...
Both these things are granting the exact same ability, the ability to move some distance, to the character, just under different conditions. There is no difference either specified or implied between the way in which either ability says "you can move" that would differentiate it from the other.
Even the language used supports the fact that you are able to "spend movement" with the movement granted to you by skirmisher reaction.
Freedom of movement says:
spend 5 feet of movement
The general rule on speed says:
Your movement...
And skirmisher reaction says:
This movement...
If even the description says all of these are "movement", why can you not spend them the same way: as movement?
If you do not allow the character to "spend movement" with skirmisher reaction then they also would not be able to stand up from prone, move while prone, or move accross difficult terrain in any way. None of which makes sense, since, again, there is no essential difference between the movement granted on your turn and the movement granted by skirmisher reaction.
In 5e there is no such thing as a move action. You simply have [your speed] amount of movement that you can spend however you are able. Normally, you can only move during your turn and you have 0 movement outside of it. This is an exception, giving you [your speed/2] movement that you can use as a reaction during someone else's turn.
Naturally you can spend your movement however you like (moving, jumping, standing up, using freedom of movement, etc.).
If you couldn't use movement with the Skirmisher feature and instead only move, nothing would stop you from moving while tied up and lying down, likewise you wouldn't be able to climb or jump as part of your movement, rather un-typically move in some neutral fashion to your new location ignoring reality around you.
Another point in favour of this view is the ready action. By spending an action and selecting a trigger you can "choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it". This is another case where you are allowed to move, but it is not specifically spelled out that it gives you movement. Like in the Skirmisher ability, it would be nonsensical to disallow you to climb, swim, etc. during your readied movement.
Yes.
Movement is movement. If you have the ability to move 5' or more at any given time, then you have 5' of movement to spend. Saying "no" is like saying that the Rogue Skirmisher Reaction would not allow you to jump, or climb, or swim, but only to walk.
No - spending move and moving are not the same thing.
Freedom of Movement is allowing you to spend 5 feet of movement in order to escape a grapple. You are not actually moving anywhere (you stay in the same space), and there is no indication that you would be allowed to perform this escape when it was not your turn.
The Rogue Scout ability, by contrast, is spending your reaction to move - and you are moving an amount equal to half your usual move, without actually spending any of your move for that turn.
