What's "being named" mean here?

What does "Share A to B" mean here?

In this context, to "share A to B" means to relate to someone on something or to agree with them on something. Ex: I share your desire to help the world. Ex: His daughter shares his love of music with him.

Can "funny" be used in this context without changing the meaning?

No, it would change the meaning of his response! Here he's being sarcastic by saying "sounds fun" since it doesn't actually sound "fun." If he said "sounds funny," it would mean he found it humorous and wouldn't have the same effect that sarcasm has. Ex: Swimming with dolphins sounds fun. = seems like a fun, enjoyable activity = Swimming with dolphins sounds funny. = seems humorous to do Ex: The name of the movie sounds fun. = seems like an enjoyable movie = The name of the movie sounds funny. = seems like a humorous title.

Could you explain the difference between "vegetarian" and "vegan"?

These are both dietary lifestyles. You can consider "veganism" to be like an extreme form of "vegetarianism." Vegetarians do not eat meat but do consume seafood and other animal byproducts like eggs, milk, and honey. Vegans do not consume any food that comes from animals at all, so they only eat vegetables, fruits, nuts, grains, and root vegetables like potatoes.

"Like you do" and "the way you do", What's different between these two? 

There is no major difference between "like you do" and "the way you do" in this case. The only difference is that "like" is used more often when making comparisons, and can be used more frequently in lots of different contexts. The phrase "the way" in "the way you do" refers more specifically to the style in which someone does something, and so it means in "the manner" or "how" you do. However, this is only a slight difference and these phrases can often be interchanged. Ex: I need your help on the homework because I don't understand math like you do. Ex: I'm shy. I don't make friends the way you do.

Are there any nuances that this "across" specifically implies? Does "across" here imply "one after another"? What is the difference between "across the country" and "in the whole country"?

Yes, there is a specific nuance expressed with "across" here. "Across the country" in this context implies that clinics are widespread or scattered out evenly in the country, where as "in the whole country" just expresses a total amount. So "in the whole country" could mean that something is clumped in the same area of the country while "across" indicates that they are evenly distributed. Ex: There are several lights across the room. Ex: There are several lights in the room. In this example, the first sentence indicates that the lights are spread out in the room whereas the second sentence implies that they could be close together in one area.