What does "gotcha" mean?

What does "ease anger" mean? Is it common to use the verb "ease" with emotions?

To "ease" means to soothe, alleviate, or reduce the severity of something. So to "ease anger" means to make the feeling of anger less strong. The word "ease" is normally related to feelings of physical pain but it can also be used for emotions which may cause suffering. Ex: The injured basketball player put ice on his foot to ease the swelling. Ex: She made me a cup of tea to ease my sore throat. Ex: I got a massage to ease my stress.

What does the word "wave" used in this context mean?

The word "wave" in this context is similar to stage or step, with heavy force. The first "wave" would be their first invasion. Then maybe they'd drawback and get ready to invade again for the second "wave". Ex: We had a sale today. The first wave of customers came in at lunchtime. The second wave came in the afternoon. Ex: A wave of happiness suddenly came over me. Then I was sad again.

Could you tell us the nuance of this expression? How can I use it in other situation?

Here, "I'll tell you what" is an informal expression that is used to introduce a suggestion. Chandler says "I'll tell you what" and then suggests that he and Joey spend more time together. Ex: I'll tell you what. Why don't we start working out together? That way I can help train you. It is also used emphasize a statement that will follow. Ex: I'll tell you what. He seems strict but he's actually a really fun guy.

Is the phrase "smaller than average" used as an adjective here? also, is it okay to say "might provide dinner plates which are smaller than average"?

Yes, "smaller than average" here is used as an adjective. The rule is generally to hyphenate words which are put together to create an adjective which describe a noun that comes directly after the adjective. So it is actually more correct to use "smaller-than-average" here with hyphenation. It is also correct to say "might provide dinner plates which are smaller than average", and in this case, you do not need to hyphenate "smaller than average" since it comes after the noun in the sentence.

I don't know well about "out there". Isn't it okay to use only "they're somewhere" without "out there"?

"Out there" here means "somewhere beyond my immediate surroundings". Just "somewhere" implies someone or something can be anywhere, including near you. Therefore, while saying "they're somewhere" is not wrong, saying "they're out there somewhere" is more specific and better to say. Ex: Dangerous animals are out there. (Meaning dangerous animals are not near you, but somewhere away from your location)