War Caster lets you cast a spell instead of an opportunity attack, which is made just before the creature leaves your reach. If your reach is 5 feet and you use War Caster to cast a ranged spell attack then, yes, the attack is made with disadvantage.
If you have a natural reach greater than 5 feet, then the attack is made when the creature is more than 5 feet away, so this doesn't apply. Similarly, it doesn't apply to melee spell attacks or spells that use saving throws. You are also good if you have the Crossbow Expert feat.
You can cast still spells without free hands
That is exactly the purpose of the Still Spell metamagic feat, it removes the somatic component of spellcasting.
Somatic (S)
A somatic component is a measured and precise movement of the hand. You must have at least one hand free to provide a somatic component.
But you cannot cast most spells while pinned
Yes, even still spells might be impossible while pinned, as the conditions say:
A pinned creature can take verbal and mental actions, but cannot cast any spells that require a somatic or material component.
This means you require a spell without somatic components and without material components. So most spells are out, unless you have both Still Spell and Eschew Materials.
Material (M)
A material component consists of one or more physical substances or objects that are annihilated by the spell energies in the casting process. Unless a cost is given for a material component, the cost is negligible. Don’t bother to keep track of material components with negligible cost. Assume you have all you need as long as you have your spell component pouch.
If you use the advanced spell search, only 109 spells out of 1973 (~5%) have both conditions (no somatic and no material components) without using the two mentioned feats.
Can character use his eyes for the ranged touch attack of a spell?
Yes, the spell still has a verbal component though. But he is not really using his eyes for anything other than looking at his target and aiming the spell.
What if the creature has no hands?
They can conjure spells just fine, using other parts of their body instead of their hands, as seen on this question: Can a naga cast spells even though it has no hands?
Best Answer
Here are four options:
Wait for them to spend their reaction first, or cause it to be spent on something or someone else before casting.
Use the reach metamagic.
If you have a familiar, use Spell Delivery to deliver the spell.
Some conditions can prevent your being targeted by the ability, such as giving the Fighter the unconscious condition, or you having the undetected condition.
Unfortunately although the Eschew Materials feat removes the need for materials that don't have a cost, it doesn't actually remove the Material trait, so wouldn't work to prevent Attacks of Opportunity on spells that would otherwise only gain the manipulate trait via their Material component.