Magical and magical effects are rendered ineffective in an antimagic field spell.

The lich's Paralyzing Touch is described as a Melee Spell Attack. Disrupt Life and Frightening Gaze are listed as needing saving throws against "this magic." Presumably all three of these attacks would be negated as well as any of the lich's spells.

What amount and type of damage is left that a lich could inflict if within an antimagic field spell? (e.g. such as the damage of their unarmed attack?)

Nothing built-in to the stat block

There isn't anything inherently in the Lich's *(MM, 325) stat block that gives them a non-magical attack.

But they aren't without options. The DM can give them a melee weapon (or they can take an unarmed strike - although that's just for 1 damage so that's not optimal by any means) if they really want it, but none of that does much damage and is worth their action.

The easiest 'counter' to be within an Anti-magic field is take the opportunity attack as they move out of it and then they can use any of the options they have (which a you've noted are all magical.) Once out, they can use their regular fully array of actions to deal damage, get away, end the anti-magic field or anything else that gives value to the Lich.

Lair actions are also off the table as a means to get around the AMF(my emphasis):

the lich can take a lair action to cause one of the following magical effects...

This is not a job for the lich

As user TimothyAWiseman wisely points out, this is not the best of the Lich. The Lich is highly unlikely to be alone. Minions may abound and be used to help protect the Lich from a caster getting close enough, or dealing with the caster to try and break their concentration. This is the perfect opportunity for those minions to come in and protect their master/mistress.

Assuming the Lich isn't caught by surprise by the adventurers (and most Liches' lairs will be sure to prevent that), the Lich should have a weapon of some sort handy. Anti-magic field requires concentration to maintain. Stab that spellcaster. Realistically, expecting incoming adventurers, the Lich should also have minions of some sort (probably lesser undead) to stop the PCs from getting within 10 feet in the first place, and to break their concentration on their buffs and AMF.

Of course, if the AMF wasn't already running when the PCs break in on the Lich, the Lich also has counterspell and 8th level spell slots.

The Lich's lair actions, while seemingly magical in nature, seem to be part and parcel of being a Lich - calling forth spirits to attack the AMF caster or tethering them and allowing damage to the Lich to break their concentration should both also work. Also, mundane traps that are triggered by the Lich can help either move the AMF or disrupt concentration (wall of poison darts, trapdoor, etc).