Would it sound awkward if I exchange the word "believe" and "expectation", and say "he expected that this time his beliefs would align with reality" instead?

What does "all the time" mean?

"All the time" simply means "frequently". Ex: I love watching movies, I watch them all the time. Ex: I eat apples all the time to stay healthy.

What is "without further ado"?

"Without further ado" is an idiom that means "without waiting any longer". This idiom is often used when introducing someone or something to an audience. Ex: Without further ado, I present to you the group you've all been waiting for - BTS!

What does “be after someone” mean?

"Be after someone" means to look for someone. When someone is after you, it means that someone looking for you. In this case, it is used when you are being chased by someone. "After something" basically means to look for a particular thing. Here are some examples of "be after someone/something": Ex: That guy just robbed a bank and the police are after him. Ex: I'm after a particular sauce that goes well with vegetarian dish.

Should I use "man" to only male?

Yes, you should use only "man" when speaking casually to a male friend or acquaintance. Alternatively, you can use "dude" or "bro" with similar effect. Ex: Thanks for the beer, man. Ex: Hey man, thanks for lending me your car today.

Can I say “pool of water” instead of “puddle of water”? Is there any other way to say “puddle of water”? 😇

Yes, you can say "pool of water" here instead of "puddle of water". A "puddle" however, implies a smaller amount of water than a "pool". In this context you could also say a "stream of water" or say instead that the cow was "caught in a flood". Ex: The duck is swimming in a pool of water. Ex: The child is splashing around in a puddle of water.