Why did he use "catch"? Does the meaning become different if he says "see"?

What does "like them" mean here? Could you explain why did he say it twice?

He repeats "like them" here because he is emphasizing this expression to understand if Rachel actually likes them. So '"like them" like them' is used here to mean she truly likes them, because he thinks she might only be saying she likes them just because they are expensive. You can use this way of repeating "like" to clarify whether someone truly likes something or they just saying they like it but don't actually mean it. Ex: Do you "like me" like me, or just like me as a friend?

Can I say "in my life" instead of "with my life" here?

No, it would not sound as natural to say "in my life" here instead of "with my life". This is because "with" indicates use in this context. He is talking about what he wants to do "in life", however, he is expressing also what he should do with his past life experiences. Thus, he uses the preposition "with" to express the use of his life.

Is it common to consider the things as "he"?

Some people do assign gender to objects. This is most common with cars. This is because they feel a close connection to it. But it's not common for a lot of inanimate objects like a carpet. However, here the "carpet" is considered less as an object and more of a pet. Even the line "he's not gonna bite" is similar to what one would say about a dog! So using "he" is appropriate within this context. Ex: She's a beauty! What model of a car is she? Ex: Your dog is so cute. What's his name?

What's "religiously" mean here? Does it have to do something with religion?

Here, the meaning is actually non-religious! "Religiously" means consistently and regularly. So he's saying that he consistently avoids stating a political opinion. Ex: Jane goes to that cafe religiously every Saturday morning. Ex: Every month, our family goes on a road trip, almost religiously.

What's "for what it's worth" mean?

"For what it's worth" is an informal expression used to preface an opinion. It expresses that the speaker isn't sure whether their opinion will be useful or valuable, but they will say it anyway. Ex: For what it's worth, I think you're an amazing artist. Ex: I loved the song you wrote, for what it's worth.