What does "at the same time" mean?

Can I skip "anyway" here? The meaning will change?

Yes, the meaning would change slightly if you removed anyway in this case. "Anyway" here is used to show that something would still happen despite any different circumstances. Ex: Even if you could have given me a ride, I would have been late anyways.

Is it common to put "on" after the "Information"?

Yes! It is fairly common in English to put "on" after the word "information." When "on" is used after the word "information" it refers to specific facts, information, ideas, etc. about that particular subject. In this video the news anchor is discussing the statistics that are known about the Coronavirus. Ex: She received information on her birth parents. Ex: He loves learning new information on planes. Ex: I am learning information on my family lineage.

What does the "rough patch" mean in this context? In what circumstances could I use it?

A "rough patch" refers to a difficult or troubling period; a time of hardship. Here it means that in their relationship, they were having a period of difficulty together. You can use it to refer to a difficult point of time in business, a relationship, a period of life, and so on! Ex: The business went through a rough patch, and we had to close a couple of our branches. Ex: Jill and I are going through a rough patch right now, so we're going to try couple's therapy.

I've never seen a sentence starting with "should". Could you explain this a little more and maybe give me some examples?

Good point, it isn't often taught as such. Here "should" means "if," and it expresses a possible event or situation! So it's often used in a sub-clause. It's a little formal in tone, more formal than "if!" Ex: Should you go to the shops, please get me a drink. Ex: Take your umbrella should you go to the beach. It's hot outside. Ex: Should you have any complaints, you can speak to your supervisor. Ex: Write to me should you go to boarding school.

When can I use "dude"?

You can use the word "dude" when speaking with friends or people you are close to. "Dude" is a slang English word for a man; a guy. However, this does not have to be gender-specific. Some women will call her female friends dudes. I recommend you only use this with people you are close to. If you use it with someone who is older than you, he or she might take offense. It isn't an offensive word, but it is not very formal with elderly people nor is it formal with people you don't know well. Ex: Dude, where is my car? Ex: Hey dude! What are you up to? Ex: Dude, this sucks! I didn't want to fail the class.