Re: How do I use the "Declaration of Intention" to find the actual naturalization papers? #records
You indicate Sam is your brick wall and you want to find his death record and burial. I presume this is to go back in time. I too could not find Sam after the 1930 census.
It would seem that this is Sam's 9 Jun 1924 marriage to Ceil Hahn: https://www.familysearch.org/ark:/61903/1:1:2473-94J
It indicates his parents were Pesach Katz and Freida Greenberg.
Why do I think this might be his record and his parents?
The 1922 declaration of intent you supplied indicates he arrived via the Finland and that he was born in Bucharest. It does mention a "blank" day in August 1906 for his arrival. The attached is the 1 Aug 1906 ship manifest, and on line 18 there is a Samuel Katz.
Also, paging forward in the Ancestry version of the declaration of intent book does show the attached 1925 court notice, giving his address as 366 South 1st Street, not the 66 Rivington address. This matches the 1 Jun 1925 census for a Sam and Ceil Katz, off by one house number, at 364 South 1st Street, per attached, bottom left, rows 47-50, showing Sam from Romania, Ceil, and twins. Strangely, there is only a 7 May 1925 Manhattan birth record for Lila, but not Leonard, as per attached. Confusingly, perhaps the Lila listed in the attached, is NOT the Lila born to Brooklyn parents Sam and Ceil, but rather the Manhattan parents of Nathan and Ida Katz (https://www.familysearch.org/ark:/61903/1:1:X42H-N4Q).
Based on the detail of Sam's parents, this is the 1915 census record, which in Ancestry can be found here:
The 1914 marriage of Sam's brother Abraham (listed with him on the ship manifest) shows his parents as the same Pesach and Freida. See https://www.familysearch.org/ark:/61903/1:1:2431-1FZ
This Pesach passed in 1919. See either
Pesach's parent were Shabsi (Shopse) and Anna. You can locate the exact plot here: http://www.mountzioncemetery.com/search.asp
Good luck and stay safe,