Sounds like a plan
Yes, you can make the firebox from standard old red bricks. Firebricks would last longer, but are not essential for a short-term build. Especially for a short-term build, fireclay slip is probably the best mortar, as it can be knocked off leaving clean bricks when you want to build something new. It may or may not meet building code where you are; it is a bedding/sealing agent only and not a cement. (US "International Building Code" requires refractory cement mortar which is strong and sticks to bricks, though I don't know how often that is actually used.)
It sounds like you have a good tall interior chimney which should have good draft. I hope both houses do not share one flue, as that might cause complications with draft.