Exclusive Or
The rules on the ready action read:
First, you decide what perceivable circumstance will trigger your reaction. Then, 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.
In common English, "or" used in this sense is an XOR. You can you have soup or salad but not both. So, you can either move, or you can take an action.
Designer's Intent
A tweet by rules designer Jeremy Crawford agrees with this reading:
When you use Ready, you prepare an action or a move, not both.
So, in the case of a familiar, a reaction spent channeling a spell excludes it from moving away from the target.
Ready the Action for the Familiar's Turn
If you Ready the Touch spell on your turn for the trigger "When my familiar moves in range...", the familiar can use its reaction during its turn (as opposed to its master's turn). If it is close enough it could continue its movement after the reaction was complete. It would also have it's action, which it could use for things like the Help action to distract one of the enemies.
Yes, familiar can be used for gaining advantage for allies.
From Find Familiar spell description (PHB, p240), emphasis mine:
Your familiar acts independently of you, but it always
obeys your commands. In combat, it rolls its own
initiative and acts on its own turn. A familiar can't
attack, but it can take other actions as normal.
Help is listed as one of the actions that could be taken in combat and is a legitimate option for a familiar.
And it's also logical thing from roleplay point of view. From Help action description:
Alternatively, you can aid a friendly creature in
attacking a creature within 5 feet of you. You feint,
distract the target, or in some other way team up to
make your ally's attack more effective. if your ally
attacks the target before your next turn, the first attack
roll is made with advantage.
As you pointed out, even small familiars can distract enemy, which will help ally in their attack.
The Word of Designer also supports this. Jeremy Crawford answered this question in the September 2016 Sage Advice column:
Can the familiar you conjure with the find familiar spell use the Help action to grant you advantage on your attack roll? A familiar can’t attack, but it can take non-attack actions, including Help. As the text of the Help action indicates (PH, 192), the action doesn’t require you to be able to attack; you simply need to be able to provide some sort of distraction.
Best Answer
From the Find Familiar entry:
A familiar is an independent creature with its own initiative, movement, actions, and so on. So it can take an action, a bonus action, and a reaction. (Assuming it has something to do with them.)