Yes, all Katyusha rockets are from Russia. The
C-802 "Silkworm" that hit the Israeli ship was made in China. The weapons flow is all Rus/China to Iran, Syria then Hezbollah.....
"Are all Katyusha rockets from Russia?"
"Katyusha" is used as a general term these days, like all facial tissue is a "Kleenex", acetaminophen is "Tylenol", etc.
Unguided bombardment rockets have been around a long time. Even Congreve ("the rockets red glare") is a johnny-come-lately. The Chinese used them in the 12th Century.
Solid fuel military steel tubing cased extruded nitrocellulose propellant grain military rocket technology was developed in the 1920's and '30s. Various countries built weapon systems based on them. The British developed anti-aircraft weapons, America various bombardment and ground attack systems (not put in service, US military in those days had no money) and Germany deployed the famous Nebelwerfer. Japan built them and used them against us in WWII. The Russian bombardment rockets, generically called "katyusha", were more intelligent and practical designs than the Nebelwerfer upon which they were based.
Nowadays any outfit that can make "gunpowder", that is, small arms and artillery propellants, can make "katyusha" style bombardment rockets.
Fireworks and adhesive chemicals can get the job done on a hobbyist level.
Every military in the world worthy of the name uses them and they are all domestically produced. Syria appears to make them and Iran certainly does. North Korea makes them. Probably Indonesia. Pakistan and India. Egypt. Romania. Serbia. The list goes on. Chinese bombardment rockets are probably the best buy at this time but are only sold to client states.