religious claims

As an academic, if you go about trying to prove or disprove the religious claims of any religion through history, you're gonna have a bad time. Serious religious historians aren't especially interested in the validity of religious claims--they're more interested in religion as a motivating force and as a worldview.

When Mormonism was founded in 1830, the officers of the entire church consisted of a "first elder," a "second elder," and a historian. They--well, "we"--(full disclosure, I'm a practicing Mormon) have been conscious of their history from the start. As a result, Mormonism actually has an immense amount of historical source materials collected. Just off the top of my head in no particular order, here are a few of the biggest collections of early LDS materials available online (if I was talking to you in person, I could bend your ear for a long time about all the stuff not yet available online):

http://josephsmithpapers.org/

https://archive.org/details/churchhistorylibrary

http://www.sos.mo.gov/archives/resources/mormon.asp

https://eadview.lds.org/findingaid/CR%201234%201/ (Brigham Young Office Files)

http://www.latterdaytruth.org/

http://www.boap.org/LDS/Early-Saints/

https://lib.byu.edu/collections/mormon-publications-19th-20th-centuries/

https://history.lds.org/overlandtravels/

http://www.sidneyrigdon.com/dbroadhu/artindex.htm

Mormon history is immensely polarizing. As I think you've already figured from the question you're asking, Mormonism as a religion relies heavily on the actual truth of its historical claims (Did Joseph Smith its founder find an ancient record on gold plates or not? Did he talk with heavenly messengers, or not? Was Joseph Smith a clever con man who knew what he was doing, a deluded man who believed his own lies, or a prophet of God? etc.). If its historical claims are false, then the church doesn't really have a leg to stand on. So this means that there are a lot of folks who are heavily invested in "proving" or "disproving" Mormonism through defending or attacking its historical claims. I have never seen another subfield of history that can be so emotionally charged as Mormon studies.

But the field is getting better. Mormon history of 40 years ago, or honestly even 10 years ago is definitely not the same as Mormon history today. (The religious axe grinding of yesteryear is getting less and less acceptable.) Here are some highlights of recentish Mormon historiography that's coming out of some major academic presses (again, just off the top of my head in no particular order):

Ronald W. Walker et al., Massacre at Mountain Meadows (Oxford University Press)

John Turner, Brigham Young: Pioneer Prophet (Belknap Press--imprint of Harvard University)

W. Paul Reeve, Religion of a Different Color: Race and the Mormon Struggle for Whiteness (Oxford University Press)

J. Spencer Fluhman "A Peculiar People": Anti-Mormonism and the Making of Religion in Nineteenth-Century America (Univ. of North Carolina Press)

J.B. Haws, The Mormon Image in the American Mind: Fifty Years of Public Perception (Oxford University Press)

David J. Howlett, Kirtland Temple: The Biography of a Shared Mormon Space (University of Illinois Press)

Kathryn M. Daynes, More Wives than One: Transformation of the Mormon Marriage System (University of Illinois Press)

Sarah Barringer Gordon, The Mormon Question: Polygamy and Constitutional Conflict in Nineteenth Century America (University of North Carolina Press)

I'm not claiming to be making a representative sweep of Mormon studies with my list above--the field is too big!--my list reflects my own bookshelf more than anything. My point is that Mormonism is a lively subfield of academic, peer-reviewed history and not just the realm of religious axe-grinders.

When it comes down to it, if you looked at the footnotes of 2 of the most opposite books about some aspect of early Mormon history--one pro-Mormon and the other anti-Mormon--and stripped out everything modern from their footnotes so all you were left with were contemporary historical sources, they'd be identical. The polarizing aspect of Mormon history isn't in the source material, it's in the interpretation of those sources.

What about the claim that Smith was convicted of fraud in a NY court, the year before he allegedly found the tablets?

Is it true or not?

If true, doesn't it cast serious doubt on the religion?

So...i'd suggest posting this question on r/exmormon to see what you get. The biggest, and pretty darn good, response you got was from a Mormon who rightly disclosed that fact. I'm an exmo but I don't study the veracity of the religion's claims whereas many in r/exmormon do. If you want to be super balanced then give r/lds a whirl.