Does disposal order prevent me from playing Rhasa after from Last Rites after already paying it cost?

It depends on timing. Disposal Order stops Rhasa if it's played in response to the Last Rites trigger — but once the trigger has started resolving and you've begun playing (or paid for) Rhasa, it can't touch it.

• Before the trigger resolves: Last Rites goes on the Chain, and your opponent can react with Disposal Order (it has Reaction). It resolves first, recycling Rhasa from your trash to the bottom of your deck. When Last Rites resolves, you look for a unit in your trash and find Rhasa gone — you can't play it. Paying Last Rites' equip cost earlier doesn't change this; the equip and the trigger's play are separate effects (same principle as FAQ #11060 with Fizz).

• After the trigger starts resolving: no. Step 1 of playing Rhasa moves it from your trash onto the Chain (Rule 354), so it's no longer in the trash for Disposal Order to choose, and no player gets priority mid-resolution to slip a Reaction in. Once its cost is paid, Rhasa is safely on the Chain and will resolve — Disposal Order only affects cards still in trashes.