This will be an ENORMOus challenge if you really plan to make long trips-rules differ per country and also per type of train and the needed REServations will often be only able to make on counters in THAT country-with a fair chance its fully booked. Or not available at all. Re-consider. Rentging a bike for the day it well possible in many spots.

Have you searched for "bike" on the community? There are several posts regarding this, like this one.

Unless your bike is possible to fold together, it will be very tricky for you to bring your bike with you.

In countries like Sweden it's in impossible to travel with an unfolded bike on train. 

In Switzerland it is much easier as many regional trains have special compartments where you can travel with your bike, as long as you have a ticket for your bike.

Regarding the specific train from Strasbourg to Lyon you can call SNCF  (the French Railway) and ask them for help. 

https://www.sncf.com/en/customer-service/contact-us/telephone

 Press #85 for English, no booking fees, reservations are sent via e-mail.

I know but the thing here is that renting a bike is very expensive, for the days I’m travelling. Most of the times I’ll disarm my bike but I want to make a video about carrying a fully assembled bike in France an that’s the trip that best suit to me

I’ve already ask them for help and they can’t give me what I want… for now I’m trying to contact Eurail.com but they haven’t answered

Thanks! and I’ll disarm my bike most of the times but I wanna go in a train with my fully assembled bike just for whim (wanna make a video about it)

Eurail can't tell more about bike reservations than what we write here.

Could you please explain what SNCF said exactly at the phone? What was the reason they did not help you? They definitely CAN book Eurail reservations by phone and book bike reservations but it seems some of their staff are unhelpful for some reason.

Eurail can´t help you much with your bike as the local rules apply this means the rules of SNCF :/