Can I reuse old electrical wire?
You can reuse old cable if a physical inspection reveals all of it to be in good condition.  It's inside walls, so it's not like it's had 30 years of sunlight baking it.   However if it's roached anywhere, into the trash cash-for-scrap bin it goes. Shiny copper is worth over $1/pound. 

However, you can't splice except at a junction box which must remain directly accessible.  It can be behind a cabinet door that opens, but it can't be under a screwed-down plywood access panel or anything that would require tools or destruction to get to.  (the junction box cover doesn't apply, obviously). 

I for one don't have any problems with splices properly executed in a grounded steel junction box.