Da Vinci's drains
As remarked in a page of Notes and Queries (I found this at Wikisource, but I'm not sure exactly what document it's a page from!), this simply refers to the fact that Leonardo da Vinci was a famous hydraulic engineer.
In particular, he worked on draining the Pontine Marshes and made a plan to divert the River Arno. Although now most famous as an artist and scientist, in his own lifetime he found employment largely as an engineer and architect. From a recent article commemorating the 500th anniversary of his death:
in a CV he once submitted to Ludovico Forza, the Duke of Milan, he first listed his skills in bridge-building, trench drainage, weapons manufacture, and many other disciplines useful to war, before adding a throwaway mention, down at bullet point 10, of his abilities to sculpt and paint “as well as any other”.
The Pontine Marshes are far from Florence, and it seems that da Vinci's plans for the River Arno through Florence were not enacted. However, given that da Vinci spent much of his life in Florence, it seems reasonable that one might go there to study his works. Even though the museum at his birthplace was only opened in the late 20th century, various records relating to him may well have been kept in Florence.
The unpopular but necessary tax
This has been interpreted to be a reference to income tax, a hot topic of political debate in England during the mid 19th century. From "Fictions of Reciprocity in Jane Eyre and Aurora Leigh", Chapter 2 in Jill Rappoport's Giving Women: Alliance and Exchange in Victorian Culture:
Aurora's gifts have larger legal and economic significance. Just as Jane Eyre stages a debate about inheritance through Jane's recognition of a common duty, Aurora Leigh invokes a similar ethos when Aurora uses her own means of subsistence to benefit the larger community she claims as her "own". It also invokes a similar law. Income, like inheritance, was increasingly taxed throughout the nineteenth century. Whereas inheritance tax was accepted as the lesser of the two evils and even praised for its relative "unburthensomeness", income tax, lifted after 1816 but reinstated by Peel in 1842 and increased in 1854 during the Crimean War, was extremely unpoular, seen by political theorists and Household Words editorials alike as an "objectionable" measure. Despite the tax's general unpopularity, it was framed as "necessary for the public interest". It seems plausible that this is the subject of Aurora's father's speculations, when, wandering through Florence before meeting her mother, "[h]e mus[es] somewhat absently perhaps / Some English question . . whether men should pay / The unpopular but necessary tax / With left or right hand" (I: 73-76). Aurora's decision to share her home and her wealth echoes but also exceeds the income tax's "unpopular but necessary" duty to benefit the "public interest".
Left or right hand
This has been interpreted, as Peter Shor already noted, as a reference to Matthew 6:3. From Michele C Martinez, Elizabeth Barrett Browning's 'Aurora Leigh' (with commentary):
Line 76. An allusion to Matthew 6:3: 'But when thou doest alms, let not thy left hand know what thy right hand doeth.'
I'm not clear on how this biblical reference connects with the context of the poem, though, so I'll defer to Peter Shor's explanation.