What's "partner" mean here? Is it a work partner?

I don't know when to say "Britain", "England", and "United Kingdom".

The "United Kingdom" is actually a short form for "The United Kingdom of Great Britain and Northern Ireland". It is an independent country (in the same way as France or the USA) but is made up of four smaller countries; England, Scotland, Wales, and Northern Ireland. So the "United Kingdom," is like the USA, in that the four "countries" are like states. "Britain" is not really a reference to a country; it’s a landmass. It is the largest island in the cluster of islands that make up the British Isles and contains "England", Scotland and Wales. "England", like Wales and Scotland, is often called a country but it is not an independent country. It is the largest country within the United Kingdom both by landmass and population, and its capital London is also the capital of the "United Kingdom." This is probably why "England' is sometimes used to describe the whole of the "United Kingdom". So in quick summary: The "United Kingdom" or UK – a country that includes England, Scotland, Wales, and Northern Ireland. "Britain" – an island. "England" – a dependent country within the "United Kingdom". I hope this helps.

What does she mean by saying this?

She says "you have us all beat" here because he wins for having the most songs with his name in them. This is because his name is "Baby" and there are lots of songs with the word "baby" in them.

What does "yieu" mean? A dialect or an old English for the word "you"?

Great question! "Yieu" is not a word in modern English nor Old English, it's simply being pronounced this way to be humorous and rhyme with the word "adieu." You won't hear "you" being pronounced like this in any setting (other than in this song).

Since "wait a minute" have the same meaning as "wait a second," could I shorten it to "wait a min?" Just like "wait a sec!"

Yes, absolutely! In terms of asking for someone to wait for a moment, there is no difference between using "minute" or "second." You can shorten them to "sec" or "min," but the later is used more in writing and less in conversation. So if you were texting someone casually, you could say "wait a min," but it would sound a bit weird to say that in real life. "Wait a sec," though, can be used in both situations. Remember that using "sec" and "min" is more casual. Ex: Wait a minute, I'll be right there. Ex: Can you wait a second for me?

Is "your person" the right grammar?

Yes, "on your person" is correct grammar. "On your person" refers to anything that is in your pockets, purse, bag, or anything you are holding. This term is usually used when someone is being checked by security. Ex: Do you have any weapons on your person? Ex: There can't be anything on your person when you go through the metal detector.