Learn Farsi: این درسِ دهم است.

It is formed by taking the regular number dah (ten) and adding the suffix -om. This suffix turns a number into a "Type 1" ordinal number, similar to adding "-th" in English.

Why does dahom come after the noun dars, and what is that -e sound doing there?

Ordinal numbers that end in -om act just like adjectives. In Farsi, adjectives usually come after the noun they describe and are connected to it by the Ezafe vowel -e. So, dars-e dahom uses the Ezafe just like dars-e jaleb (interesting lesson).

Could I put the number before the noun instead, like in English?

Yes, but you would need to use a different suffix. To put the number first, you add -omin to make it dahomin dars, and you drop the Ezafe. Both dars-e dahom and dahomin dars mean the exact same thing!

How would a native speaker pronounce dahom ast in everyday conversation?

In spoken Farsi, the formal verb ast (is) is almost always shortened to a simple -e sound attached to the end of the previous word. A native speaker would naturally say in dars-e dahom-e.