Learn Farsi: خریدِ امروز خیلی ارزان بود.

In English we say "today's shopping" using an apostrophe-s. In Farsi, you use the Ezafe to link the noun to its possessor or time word. The word order flips to literally mean "shopping of today" (kharid-e emrooz).

kharid looks like a verb. How is it being used as the noun "shopping" here?

kharid is the past stem of the verb kharidan (to buy). In Farsi, it is very common to use a verb's past stem entirely on its own as a noun. Here, it acts perfectly as the noun for "shopping" or "purchase".

If I remove the Ezafe and just say emrooz kharid kheyli arzan bood, does it mean the same thing?

It changes the grammar and focus slightly. Without the Ezafe, emrooz (today) becomes a time word for the whole sentence, meaning "Today, shopping was very cheap." By using the Ezafe to create kharid-e emrooz, you group them into a single noun phrase specifically meaning "today's shopping trip" or "the things bought today".

Is kheyli always placed right before the adjective it describes?

Yes, when kheyli means "very", it comes directly before the adjective it intensifies, exactly like in English. Here, it sits right before arzan (cheap) to make "very cheap".