German did not really have a unified orthography until the 18th century (even if the first dictionaries came into being around the time Händel was born), so everyone more or less wrote like they thought was right. The Lutherian bible translation worked as a kind of standard guideline, rather than a rulebook. Some of the many small German countries might have standardized administrative (which was to a great part ecclesiastical back then) writing internally, but that was not the general case.
Between the 16th and 18th century, a lot of double consonants were 'en vogue' that we don't consider necessary today, like in auff (auf), Tauffe (Taufe), Kampff (Kampf). This explains the uncommon "Täuffling". And yes, orthography of a lot of double consonants after diphthongs has changed since then. (Also note how some names and "auffm" in the "Paten" column use those duplicated consonants)
The plural for words ending on ing could be built differently back then - "Täufling" already could be considered plural. Some of this is still present in German dialects ("Das sind Zwilling" in some dialects. (*)) - So I would consider the column headers to be consistently plural, other than we would write it today (Interestingly, there is no mention of mothers anywhere in that record).
The "Paten" entry I would read as
Herr Phillip Fehrsdorff, hochfl. sächsischer Verwalter zu Langendorff,
Jungfer Anna, Herrn Georg Taustens gewesenen Pfarrers zum Giebichenstein S. nachgel. Jgfr. Tochter und
H. Zacharias Kleinhempel Amts Barbier auffm Näumarckt allhier.
Some of the abbreviated words would maybe need explanation:
"Jgfr." Jungfrau, Jungfer (so, apparently, Anna was an unmarried orphan of a former priest)
(*) There is also a funny story when the German imperial engineers during WW I had a locomotive called "Zwilling" that was constructed from two paired units. In order to avoid confusion on exactly how many machines were referred to, even in official military German a single unit of such a pair was referred to as "Illing".