Yes the Germans used unguided rockets such as the R4M and 21cm mortars underwing on certain bomber destroyer units. But IIRC, some French Nieuports were carrying rockets in WWII. I don't know if those were air to air for use against observation balloons & Zeppelins, or if they were for air to ground use.
I am pretty sure that some other guided air to air missiles such as the AIM-4 Falcon, and AA-1 Alkalai were fielded before the Sidewinder, but I am 99% certain that it was the first to see combat and score a kill in battle.
As far as the similarities between the US and German guided bombs, they both operated along the same principle- radio command using visual tracking of the tail flare. And the BAT was in a league of its own, an anti shipping weapon with a radar seeker head for terminal guidance. I have also read of TV or Electro Optical guidance being developed, but the technology for that had not matured for fielding yet. The shape of things to come.
F is for FIRE, That burns down the whole town!
U is for URANIUM... BOMBS!
N is for NO SURVIVORS...
- Plankton
LSM