Probably no, but it depends if you were ever in the US prior to becoming an LPR "in any status other than as a lawful non-immigrant". i.e. if you were ever out of status for any reason, you would need to provide a status letter.
so you don't need to register or need a SS letter because you were on H1 and F1 (both status are valid non-immigrant status). Were you on F1 or H1 when you entered USA before 26? if so, you don't have to worry about it.
Let me simplify: Only people who are ages 18 to 25 AND are US permanent resident or US citizen (and maybe "out of status" people living in usa) must register for selective service.
So you are at least 31 now? If you're 31 or older, even if you were supposed to register they can't use the failure to register by itself to deny your naturalization. For denial it has to be combined with other evidence of bad behavior. So if you are at least 31 and have no arrests or any other such criminal issues, don't worry about gathering any evidence for this.The IL asks for evidence of Selective Service. But I did not have to register for it since I became a PR at age 27.
So you are at least 31 now? If you're 31 or older, even if you were supposed to register they can't use the failure to register by itself to deny your naturalization. For denial it has to be combined with other evidence of bad behavior. So if you are at least 31 and have no arrests or any other such criminal issues, don't worry about gathering any evidence for this.