The rules on somatic and material components say:

If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.

[...]

A spellcaster must have a hand free to access a spell's material components--or to hold a spellcasting focus--but it can be the same hand that he or she uses to perform somatic components.

So yes, the hand holding your focus can also perform the somatic components. It doesn't matter what the other hand is doing.

But, this only works if the spell has Material components. If it has Somatic but not Material, you will need a free hand to cast it (unless you have the Warcaster feat).

Maybe. But not likely RAW.

An Arcane Focus is (PHB, 151):

An arcane focus is a special item—an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item— designed to channel the power of arcane spells.

A DM may allow a glass eye if it was originally made to be an Arcane Focus to work (an orb is basically an eye), but I don't think I'd allow any glass eye to be a focus.

And PHB (203)

A spellcaster must have a hand free to access these components, but it can be the same hand that he or she uses to perform somatic components.

The latter quote is an explanation of how to access the component pouch/focus, but I don't think is is prescriptive in that you must use a hand. Just that if you have to use your hand, it must be free to work like that.

Slippery Slope?

Beware the fact that usually the Arcane Focus is something held in the players hand. By removing the need to hold it, you are making it much easier to do things that might have been problematic before. Feats like Warcaster are ways to mitigate holding something in both hands (dual-wielding or shield) and still allow for Somatic components (but it doesn't resolve the Material.) If the player has no need to use a hand for material components (by use of a glass eye), then it bypasses that potential restriction.

Actual Gameplay

In a current game, I've got a Paladin and my DM allowed me to tattoo my holy symbol as well as put it on my armor. This bypassed the need for material components when casting and I don't think it's made a huge impact on gameplay.

In the glass-eye case - make it a bit of a quest to find someone to make the glass eye arcane focus. That gives it some flavor, creates some tension and game around trying to find that person, and still eventually allows them to get what they're looking for.