Definitely calculate the average damage of the effect to add to CR.
Charm is harder to calculate, so I will link to a good example of monster creation for 5th edition here.
Off the top of my head, a charm effect like the spell would give a DC 15 saving throw to avoid it. I will assume average character can pass this 60% of the time.
Assuming a 4 person party will output 40 damage a round, each extra point of AC on a creature will remove 1/20 of that damage = 2 damage.
Assuming a character will fail a DC 15 wisdom saving throw 30% of the time, this equates to a party damage reduction of 0.3 * 10 damage per round = 3 damage per round. Given the damage to AC equivalency of 2 damage to 1 AC we get the Charm ability equal to an extra 1.5 AC
My assumptions on average damage per round and average DC 15 failure are wrong, most likely, but this illustrates the calculation required at least.
Going off the phrasing of the effect
When the Lich takes damage
Not "When the Lich is hit with an attack/spell/etc," when they take damage.
In this case, a Lich's resistances and immunities are applied before you determine whether or not Tether fires off. Because until you apply those, the Lich hasn't taken any damage yet.
So, since when you hit a Lich with a nonmagical sword they take Zero Damage, then even if you still ruled that Tether activated, then the Wizard would have to save against taking half of zero damage...so, no damage at all.
The specific wording you are referring to of
Resistance and then vulnerability are applied after all other modifiers to damage
Are all still applied before a creature actually takes any damage. It's not like a Fire-immune creature gets hit with a fireball, takes the fire damage, and them immediately heals that damage. They simply don't take any damage at all.
Best Answer
At the beginning of a combat, all participants roll initiative (d20 + Dex). Everyone ends up with a number, and take their turns from highest to lowest. Initiative count 20 (losing all ties) means that lair actions happen after any creature whose total for the initiative roll was 20, and before any creature whose initiative roll was 19.
So, for example, a combat starts.
Bob the Fighter: Dex 20, Dex mod +5, rolls 16 for initiative. His initiative count is 21.
Jim the Rogue: Dex 16, Dex mod +3, rolls 17 for initiative. His initiative count is 20.
Tom the Wizard: Dex 14, Dex mod +2, rolls 17 for initiative. His initiative count is 19.
Ginormous the Ancient Red Dragon: Dex 10, Dex mod +0, rolls 16 for initiative. His initiative count is 16.
Bob's turn happens first, followed by Jim's. Then Ginormous can use his lair action. Then Tom takes his turn, and finally Ginormous takes his.