If I'm reading this right, the caster of the antimagic field can exclude themselves and cast spells from within it on other targets. Then, other casters can target them. But they could not be fireballed, and melee inside the field would not have magic weapons. So, a lich would render themselves immune to melee damage but be able to cast spells on those outside the field. 

The lich could still attack with their weapon and use the magical properties of it. 

I really don't think that's the intention even if it might be rules as written. Where are you getting the idea that the caster can exclude themselves from the effects?

rules-glossary

Emanation [Area of Effect]

An Emanation is an area of effect that extends in straight lines from a creature or an object in all directions. The effect that creates an Emanation specifies the distance it extends.

An Emanation moves with the creature or object that is its origin unless it is an instantaneous or a stationary effect.

An Emanation’s origin (creature or object) isn’t included in the area of effect unless its creator decides otherwise.

posted below the emanation text

If I am reading correctly this is somewhat true (although I highly doubt it's intended). From the rule that you're talking about in "Emanation" you could exclude yourself from the Emanation of the spell. This kind of gets into a SUPER stupid rules lawyery discussion about where you think the Emanation starts/stops on the origin side. If it starts even a tiny bit off of the subject then any weapon would regain its abilities immediately before hitting the person, and any spell would un-suppress just in time to hit the person. You would most likely NEED to rule it like this since there is no rule (that I'm aware of) covering the start point of most offensive spells so theoretically you could argue the magic starts within you, and then is suppressed through the field and starts again immediately after. If you argued this so that the Emanation is touching, but not including you, then no magic can affect you because there isn't a point without anti-magic between you and the field. If you argue the emanation starts even like a millimeter off the person then all things would function as normal on the target but would exclude the area around them. Even Fireball could be ruled to hit the creature, even the new one without explicitly "going around corners" now operates under the Sphere Area of Effect, which means all lines of the fireball AoE will extend out and through the anti-magic sphere, hit the person in the emanation and then continue on the other side. Since the Anti-Magic Shell doesn't block the Line of the AoE but suppresses it.

So you can either rule the old way of making the person inside affected no matter what, or discuss with the player before hand what their expectations of the spell are if they want to not be affected by the Emanation.

Personally I don't understand the "Exclusion form Emanation" piece of that rule, I would have guessed it was in place to all the new Conjure spells, and things like Spirit Guardians, but I'm pretty sure any spell like that allows you to choose if a creature makes a Save. Maybe I'm missing something though.

Going by the text I disagree. "Areas of effect created by spells or other magic can’t extend into the aura," so fireball wold stop outside of the field. As far as what is excluded, it says the object or creature therefore no fireball could start in in it since it occupied space.  Weapons wouldn't work because they are never in the creature (since in this example it was a lich immune to non magical damage) 

Some effects travel others don't so I'd rule some spells would target fine others not it would depend. 

Rule it however you want. I said COULD be ruled to hit the creature. I would never let the fireball hit something in an anti-magic shell cause I would never use this interpretation of avoiding the emanation in my game. I am just saying that if you DO use it, there are a TON of "gotcha" considerations that any sort of bad faith player or DM could use, so you will end up having to consider how every one of your spells actually functions.
Do you want your fireballs to act like a burst of fire, or an area that turns into a ball of fire, do you want light to emanate out from a point like a natural light, or magically illuminate a designated space. When you cast cloud kill can you just hold your breath cause the spell can't get into your lungs, or does the spell occupy the entire area and is already in your lungs. There are tons of semantic questions the game doesn't answer, but HAS answered in various ways in the past, so different players might have different expectations of how things work. Just make it clear how you expect this interaction to work if you're going to use the current Emanation rules verbatim.

the 2024 Version of Druid Wild Shape no longer says it is a magical effect. So.. given that would a druid be able to change or keep a transformation of wild shape in an anti-magic field?

From the class description in 2024 PHB:

"Druids belong to ancient orders that call on the forces of nature. Harnessing the magic of animals, plants, and the four elements, Druids heal, transform into animals, and wield elemental destruction."

That lists the transformation into animals as magical, so I would still have it affected. 

Gotcha, it just seems super straightforward, RAW. I guess I don't do bad-faith players, so I'm less attuned to that. What I'm talking about are the clear ways the rules are written not odd circumstances that are outside the game's scope. No room at the table for that nonsense.