This is short for the longer phrase "vamos a ver" which means "let's see". But in this context it is used to get someone's attention and is similar to the English phrase "come on" or "listen up".

The phrase "que cara de ..." is useful to describe someone's feelings or to comment on how you think someone is feeling.

Again, the diminutive of the word is used to be more informal and friendly. This change in form alters the meaning from cafe or "a cup of coffee" to cafelito which is more like "a quick cuppa".

There are many options when asking for the bill. It might be more common to say "me cobras?" or "cóbrate" in an informal setting at a bar while keeping the more reserved "la factura, por favor" for a meal in a restaurant.

This can seem quite abrupt if directly translated into English: "put me a coffee". But it is not considered impolite. In fact, directly translating a more polite English structure into Spanish (e.g. I would like a coffee -> quisiera un cafe) might be seen as affected and maybe even sarcastic to a native Spanish speaker. If you use ponme with a por favor and a smile then that's as polite as you need to be.

This is the diminutive of the word copa which gives it less formality. You would use copita instead of copa if you want to encourage or convince someone to have a drink by implying it will be small and quick. In English you would say "a swift one" or  "a quick drinky"