Seat reservations are totally separate from the app, even those made on interrail.eu won't appear there.

Whenever asked on the train show both the app (your ticket) and the PDF/print (seat reservation).

Is it possible to make the reserve on the original websites?
For example, I need to reserve From Madrid to Toledo, but the App shows no available trains by 27th December. 

Most of the time yes you can and it's a great way to avoid fees. Unfortunately RENFE is a real pain. Their website isn't user-friendly and you can't book Interrail/Eurail reservations there. For a few years it was impossible to book Spanish trains online at all (RENFE discontinued that option for no reason) !

Some trains are now available on interrail.eu/eurail.com (mostly AVEs) but for the rest you'll have to visit a ticket counter in Spain or Germany or... buy a ticket.

More info here : https://www.seat61.com/interrail-and-eurail-reservations.htm

The other issue is that not all trains after 10th December (European-wide timetable change) have been added on Rail Planner yet. RENFE is notoriously late about that, some trains are loaded on Rail Planner a week or two before departure.

EDIT : I had a look and looks like tickets are not even on sale on renfe.com right now. You'll have to wait a bit. A ticket seems to cost 13.90€ while a reservation is 4-6€ so it may be more convenient to buy a ticket for this journey.