Learn Polish: Kura je i pije. - The hen is eating and drinking.

The word je is the "he/she/it" form of the verb jeść (to eat). It is very common to confuse it with jest, which means "is". Remember that kura je means "the hen eats", while kura jest means "the hen is".

Does Polish have a different way to say "is eating" versus "eats"?

No, Polish doesn't have a separate continuous tense like English. The present tense verb je translates to both "eats" and "is eating", just like pije means both "drinks" and "is drinking". You figure out which one makes sense from the context.

Can I use kura if I want to order chicken at a restaurant?

You wouldn't use kura for food. Kura specifically refers to a live hen. If you want to talk about chicken as meat or food, you should use the word kurczak.

Why isn't there a word for "the" in front of kura?

Polish does not use articles like "a" or "the". The single word kura translates to either "a hen" or "the hen" depending entirely on what makes the most sense in your conversation.