Not to discourage discussion here, but feel free to cross post to r/linguistics as well! I’m sure they have a lot of users knowledgeable about reconstructing pronunciation even if they’re don’t have all the answers for Sumerian.
Hey, recent linguistics graduate here with some experience in Sumerian!
We have some understanding of the language - we have a rough map of it's phonemes and the range of sounds they may have been, but we're less sure about the vowels or how sounds vary in position. In terms of word order and conjugation, modern understanding outpaces the understanding of it's sound system. For instance, we know it had at least four vowels with short and long forms, a two-way sound distinction in it's plosives.
[Edit: I didn't add much about the details of phonemes, phonotactics etc, as I wasn't sure if you wanted that as well, but if your curious A Descriptive Grammar of Sumerian, Abraham Hendrik Jagersma , 1955 is a very thorough start.]
The how actually involves the script it is written in, Cuneiform - It's one of the three or four main spontaneous creations of human writing as far as we know, and was used and adapted in a variety of regional languages, including Elamite, Hittite, and Akkadian.
This writing system is sort of a hodge-podge to what you may expect: It is written in a combination of Ideogram (Individual or multi-graph glyphs map directly to a ideal word with no indication of how it's pronounced) and a Syllablery (Glyphs represent syllables and occasionally, independent sounds). These are frequently used together in context, which means distinguishing whether a glyph is a Ideogram or a syllable is difficult. Given that this was a language that was morphological very complex, with long, derived words and multiple suffixes and prefixes per word at times, in addition to unique glyph formed for formalist flourish or to try to logically extend the meaning. Additionally, it is a conservative writing system, and so glyphs may distinguish contrasting sounds that had long stopped existing, or new contrasts were not well noted, as well as general inconsistency in writing - /e/ and /i/ often varied, for instance.
The script was originally deciphered via a trilingual inscription, with Old Persian and Old Babylonian (Akkadian) among others, via knowledge of Old Persian titles. We then transliterated Akkadian, a Semitic language with enough relation to Hebrew and other well understood classical Near East languages, and with that finally leads to Sumerian.
Sumerian had a lot of contact with several parts of the ancient Near East and Mediterranean, indicated above by how many different peoples adopted the script, and one of those was Akkadian/Babylonian, which was heavy influenced by Sumerian and adopted numerous loanwords, and vice versa. As time went on, most of the population eventually stopped speaking Sumerian and instead spoke other regional languages, critically Akkadian, as Akkadian scribes continued to use the script for their own language as well as Sumerian proper as a scribal language.
It is through Akkadian that we get most of our information, as well as occasional writings in Greek and loans from Sumerian in Akkadian and Akkadian in Sumerian, and through their writings, there discussions of the language, mistakes and alternations in spelling suggestions, as well as features in the script itself, such as a lack of representation of final stops.
This sort of research via spelling mistakes, analysis of legal combination of sounds, loanwords and diachronic changes across various languages is the one most responsible for helping deciphering old languages, such as Egyptian via Greek and Coptic, or Old Chinese via rhyming dictionaries. It's the best we can do, often involving some clever leaps while researching how language change impacted earlier text, as sadly most ancient peoples do not write exhaustively accurate descriptions of their own languages often.
Did sumerian cities all speak the same language (or closely related languages) or was there a very high degree of linguistic diversity?