Learn Spanish (Latin America): Ese perro está muy flaco.

Ese is the masculine word for 'that' when it comes right before a masculine noun like perro. Eso is also a word for 'that', but it is a neutral word used for general situations or unknown objects, never directly before a noun.

Why do we use está instead of es here?

We use está (from the verb estar) because the dog's skinniness is being described as a current physical condition or state, possibly meaning he looks skinnier than usual. If we used es (from ser), it would mean the dog is just naturally a skinny breed or type of dog.

Could I say mucho flaco instead of muy flaco?

No, you need muy here. Muy translates to 'very' and is always used before adjectives like flaco (skinny). Mucho means 'a lot' or 'much', and is used with nouns (like 'a lot of food') or after verbs.

How would I use the comparative structures from this lesson to say 'That dog is skinnier than my dog'?

You would use the formula más + adjective + que. To say 'That dog is skinnier than my dog', you simply replace muy and say Ese perro está más flaco que mi perro.