The only way I can use recyclers is to choose "Harvest" command when I see debris on the galaxy map.
In that case I can not choose how many harversters to send, the system chooses the necessary quantity of the ships.
Is there a way to send more recyclers in order to speed up collection of debris?

Another issue is sending recyclers in Attack missions.
Is it possible to do this, and start recycling immediately after defeating the enemy?

yes there is a way go to your fleet option and then chose how many recyclers you want to send it is mention there and let you choose there, and for the second question when you send the recyclers in harvest mission you can add attack ships with them so that they can go in defend mode when face an enemy while recycling. 

Will you please explain where to find these "fleet options"?

When I click on debris I see only "Harvest" command, and can't specify how many recyclers to send.

The game doesn't allow me to send recyclers to specific planet coordinates - when I enter coordinates manually, no missions are available...

You should choice "coordinates" type and select some recycles ships: http://prntscr.com/c62o2a
Debris from battles are around planet (on coordinates).

Everything's clear now.
Thanks for the explanation. :)

Great