Tranny or clutch? 5th gear issue... *searched*.
 

When I enter the 5th gear and I push the gas pedal, it comes back to neutral. *No grind*. If I keep pushing the 5th gear with my hand while I push the gas pedal, it's all fine.

Sounds like a tranny problem or clutch?

I'd say tranny but I just wanna be sure...

J-Spec or rebuild? I already have an old J-Spec that make some noises but is running #1.

Defiantly tranny something is keeping it from fully engaging 5th whether its a syncro or shift fork or shift rod something is hanging it up.

As far as rebuild or the jspec sounds like they both will need a rebuild so take your pick. The jspec may last awhile or it could go tits up on in short order kinda hard to say.

Took it for a 10 kms ride. Now, my tranny feels like it's empty! I can't find any gear. Happened to anyone? I never saw this kind of problem...

Where is your clutch taking up at top,mid,or bottom of pedal? Sounds like something broke or came loose in the shift linkage some where.

Mid, clutch has 6,000 kms. I achieved to find the 4th gear, otherwise I have NO gear, I can move the shifter everywhere with 0 restriction, like if it was empty. What did I break?

I unfortunately don't know a lot about the FDs trans and my searches haven't turned up much that was useful. My guess is that it will be something in close proximity to the shifter like the shift select spindle or the end of the shifter actually being broken off.

I'd start by pulling the shifter out and seeing if you find anything that is broken or worn out looking.

It is also possible that you munched the 5th syncro and a piece went through and shredded the trans though I'd thing you would have felt that.