The thicker the wire ,the easier it will detach .Thinner wire = less vibrations
Use the wires that will come with your piezo ,glue the wires on the piezo and apply tape.
For removing solder search on ebay:
"Vacuum De-solder Gun Desoldering Pump Solder Sucker Removal Remover"
They cost less than 2$
Also when soldering with iron you
must use soldering paste which is cheap as well . Otherwise the temperature gets high and might damage the piezo
PS. Those are friendly advises and not criticism