Which sounds more natural to native speakers between "my name is..." and "I'm ..."?

What's "ease" mean here?

The way I understand it, "ease" refers to waking up - "easing into" the day, moving carefully and calmly into the day. It's not commonly used like this. I'd consider this very poetic or dated English. Ex: They ease you into horseback riding by placing you on a horse and walking you around. Ex: I ease into the day with a cup of coffee.

Which sounds more natural to native speakers between "my name is..." and "I'm ..."?

They are both acceptable as simple introductions! There is no more "natural" expression, but "I'm [name]" is a bit more casual due to its simplicity and conciseness. Make sure that you use an expression appropriate for the occasion (i.e. formal or professional vs. casual or informal). Ex: Hello, my name is Stephanie. It's great to meet you all. (More formal) Ex: Hey guys, what's up? (Casual) Ex: Hi everyone, I'm Stephanie. Happy to be here. (Professional)

Shouldn't it be "a part of work" with indefinite article "a"?

The difference between "part of" and "a part of" is subtle. In general "part of" means something that belongs to or is essential to something else. In this case, it means that going to the gym is essential to her work. While "a part of" means a piece, or a segment of something. In this case, both these terms could be used interchangeably and not affect the meaning of the sentence.

What does he mean by "that ain't doin' too much"?

To "do too much" is a slang expression which means to "overdo" something, or to do some actions that seem overboard or unnecessary. When he says "that ain't doing much," he's saying that it's a level of interaction that is just right and not too much. Ex: You bought him a Rolex AND a car for his birthday? You're doing too much. Ex: This song is trying to do too much. It sounds messy and loud.

What does "free speech" mean?

"Free speech" is the short form of "freedom of speech," which means the right to express opinions without censorship or restraint. Ex: Free speech is a human right. Ex: I heard that the website censors what people say. There's no free speech.