Is there some simple way to clean syringe and needle after each use (solder dispensing)? I was thinking to leave in hot water for a while.
Seal up your syringe, with needle attached, in a sealed bag, then put it away in your cold storage fridge.
When you get it out again to use, 95% of the time you can just push hard on the plunger and clear out the solder paste that was sitting inside the needle and new paste starts to flow. So cleaning it out isn't really needed. Only occasionally do you need to replace the needle.
Just be aware of the ball size of the paste you are using (T number) vs the size of needle you are using.
Small needles will clog a lot if you are using a paste with large ball sizes.
For general stuff like 0603. 0805, TQFP, SOIC only a single needle size is needed.
It's only when you start wanting to paste DFNs and 0402 and smaller that you start to get issues and need a smaller needle.
Manually pasting DFNs and 0402 can get a bit challenging, totally doable but takes a bit more time.
For DFNs i normally don't manually paste them, I pass a tinned iron tip over the entire footprint with some flux to get solder on all pads, then place the DFN on top with more flux and hot air it on that way, but if doing this, you want to do it after after reflowing all other parts. You can't hit the PCB with hot air while it has paste on other pads or the heat will make a mess.