We don't often say "never" in plumbing, but we can say with some certainty that water NEVER runs uphill!
What's the story on that wooden box? Does the floor drain somehow run over there? It sounds from your description that the only way you could have a toilet in the basement is with a sump and pump. ( It would be a sealed sump, not like the one your sink etc. drains to.)
Another choice is a macerating toilet, which is also pumped but needs a smaller discharge.
In either case, you are going to need a vent. Could that 2" pipe you mentioned be a vent?