To add to the comment of /u/jschooltiger I know of several other names that date from around the same period.

Due to the nature of how writing developed in Sumer that is in the context of accounting, trade and other economic interaction the first Sumerian names we have are those of slaves and accountants. We have the slave owner Gal-Sal and his two slaves Enpap-x and Sukkalgir (3200-3100 BCE). We also have Turgunu Sanga (3100 BCE), an accountant for the Turgunu family.

As for the oldest name still in use I'm not sure that question is definitely answerable without being able to access the naming conventions of every people in the world and the history thereof. Sure we can look at the the earliest Sumerian or Egyptian writing and date a particular use of a name reasonably well. However there is no way to say when say Turgunu Sanga was first used as a name. Presumably the first appearance of Turgunu Sanga was not on the slate of OIM A12259 in 3200 BCE. There is no way to definitely tell the age of the name and it could be as old as the earliest spoken Sumerian which goes considerably further back than than 3200 BCE.

Thinking a bit further it would be true nightmare to determine that age of say Yakut names when the first written Yakut is from the 20th century. In fact any people who were illiterate until relatively recently recently pose quite the problem in this situation.

So I don't really have an answer to the second part of your question I guess.

The problem with your second question is that even with the example you cited, David, we pronounce it and spell it different from the way it was spelled a long time ago. So the question arises, at what point does it change so much that it becomes a different name entirely?

If the first name still in (more or less continuous) use is from the Hebrew Bible, there are a couple of candidates. It's, however, hard to date exactly, because the dating of the Hebrew Bible is a very contentious issue.

Moses, Miriam, Deborah, and a few others are all associated Textual critics often believe the oldest unchanged parts of the Hebrew Bible are the little poems like the Song of the Sea, Miriam's song for the women, and Deborah's song in Judges 5 I believe all of which are noted for their archaic language compared to the rest of the Hebrew Bible. (Textual critics argue that the current form of the "Song of the Sea" cannot pre-date Temple; it also doesn't explicitly mention a Moses, though it's clear it's the same story told in Exodus and Joshua, though according to many textual critics, a slightly different version). The Song of Deborah is similarly hard to date, but it is consistently thought of as very old, though like much of the Hebrew Bible, there's a huge range of dates given (most 1200 BCE to about 600 BCE, but a few even arguing it's post-exhilic). And, of course, the Song of Deborah, in addition to mentioning the name Deborah and Yael and Barak, mentions several tribal names (Benjamin, Ephraim, etc) which are still in use today (though these, like Israel, weren't necessarily in continuous use as first names).

Since it's hard to date these texts with any sort of certain or consensus; for clear consensus choices, you'd have to turn to Biblical archeology if the oldest name in contemporary use is from the Hebrew Bible. The oldest Biblical names recorded on monuments outside the Bible are David and Omri on the c. 840 BCE Mesha Stele and c. 800 BCE Tel Dan Stele (for more on the historical David, which applies almost identically to the historical Omri, see this older post), and both used in such a way indicating that they are dynastic founders, thus indicating that the names are older than the stelae themselves (the name Biblical name Jehu may also be found on the Black Obelisk of Shalmaneser III around this time but, like the first reference to David, the inscription is damaged in a way to make the reading ambiguous).

As per the first half of your question, what is the oldest name we have recorded, a very similar question has been asked before.
This answer comes from user u/missingpuzzle.
https://www.reddit.com/r/AskHistorians/comments/1tqfef/what_is_the_oldest_recorded_human_name/

The oldest recorded name question has been asked here before; the answer is probably Iry-Hor, a pharaoh who lived around 3200 BC.

As far as your second question, that's going to be a tough one to answer, but we have a wide enough range of experts here that we'll see what they say.

That said, a reminder to respondents: Answers in AskHistorians are expected to be comprehensive, contextualized, with sources provided on request, and people who answer questions are expected to have an expert level knowledge of the subject. Just posting "I know a dude named David, that's an old name" will get the post removed and the poster sent to the netherworld.