Questions about the editing of Edward 321, which does not appear accurate 
I have the Spreti source in front of me, and sure look different that what you are reporting...

1)Spreti says that the family was of the old tradition (the old order of Rome) originally of the gens Catellia, of the Roman consulars?, p.274  

2)Spreti says that Johannes Barbaro, prester, with a clear autograph was capable for the concession in 982 for the church of Santo Zorzi?, p274

3)Spreti says that by 992 the Barbaro were Authorities of the Republic and were part of the patriciate at the serrata for the Great Council in 1297?, p.275 

4)Spreti says that the imperial eagle was "framing" Francesco Barbaro's arms, not quartering, which the old Italian word  looks like, p.275 

5) Spreti says that after the Congress of Vienna (Sovr. Risoluz.) which was assembled to give back what Napoleon took, the (original nobility) as Spreti states both the titles of Patrician of Venice and Nobile were authorized. 

6) Spretti says that Alessando Barbaro (1764-1839) was a member of the Council of Ten, High Counciler of the Supreme Tribunal of Verona, and owner of Palazzo Dario, p-277-288

7) Spretti also talks about the aviation achievments of Aldo Barbaro as well, p. 278

8) Spretti also adds a whole other section for the familiy's foreign titles that were granted by royal decree in the encylopedia that you seem not to mention at all...