Genealogy /
THE MEANING AND RESEARCH OF MY POLISH LAST NAME, SURNAME? [4500]
TYSZKA: This nickname-turned surname originated as the hypocoristic (pet) form of such first names as Tymoteusz or Tytus, so at least the first name given would be the equivalent of Tim or Timmy (from Timothy). Another common form is Tyszko. Most every Polish surname has been used by Jews at one time or another, but that does not make it a typically Jewish name. Typically Jewish names include Lewin, Szapiro, Margolis, Kabała and those incorporating the names of precious substances: perl, rubin, diamant, silber, gold, bursztyn, etc.
Tyska would be the mazurianised (peasant dialectal) version.
GADZIŃSKI: root-word gadzina (poisonous snake, viper, reptile); also said of a vicious, toxic person); most likely a toponymic nickname for an inhabitant of Gadzinowo (Viperton, Snakeville). You are correct that its noble line stamped their documents, adorned their manor house façades and engraved their tombs with the Pobóg heraldic device.
PAWLOSKI: This is the regional (probably Śląsk/Silesia) version of Pawłowski. The latter is a toponymic nick for someone from Pawłów or Pawłowo (Paulville, Paulton).
DZIDO: probably from dzida (spear - ancient weapon comprising a pointed blade mounted in a long thin shaft); another possibility is dzid -- an Ukrainianised word from grandfather or old man (Polish: dziad).
RĘBISZ: names with the rąb~ręb component refer to wood-cutting or forest-clearing, so it could have emerged as an occupational nick; but in some cases it may also be traceable to toponymic sources such as the villages of Rębisze, Rębiszewo and similar.