Difference between 什么都不 and 什么都没
Let's begin with 什么都 meaning &quot;everything&quot;; see Expressing &quot;everything&quot; with &quot;shenme dou&quot; at the Chinese Grammar Wiki.  We can add a verb or an adjective afterwards:
什么都 + adjective

什么都美丽 (everything is beautiful)
什么都辣 (everything is spicy)
什么都简单 (everything is simple)

which we negate with 不:

什么都不美丽 (everything is not beautiful)
什么都不辣 (everything is not spicy)
什么都不简单 (everything is not simple)

This is the easy case.  See also the &quot;Negating Adjectives&quot; section of Standard negation with &quot;bu&quot;.
什么都 + verb

什么都有 (to have everything)
什么都吃 (to eat everything)
什么都买 (to buy everything)

which we negate with 不 or 没 or 没有, which is where it gets tricky.
At this point, it's no longer about 什么都, but only about 没 vs. 不.  The most prominent Chinese.SE post about this is Why is 有 (yǒu) the only verb that requires 没 while other verbs can use 不?.  See also the Chinese Grammar Wiki's article Comparing &quot;bu&quot; and &quot;mei&quot;.
The following is a common construction; how it's usually taught.  Given tsh's comment below, it looks like there are other constructions beyond this (which I'm unable to comment on).
什么都不 + verb vs. 什么都没 + verb
Indeed &quot;没 + verb&quot; often indicates a past non-event. From Negation of past actions with &quot;meiyou&quot;:

Usually verbs can be negated with 不 (bù), but that construction is used for habitual or present actions. If the verb is about an action in the past, though, 没有 (méiyǒu) should be used.

So for example:

什么都没吃 = 什么都没有吃 (to not eat anything [in the past])
什么都不吃 (to not eat anything [present or future])
什么都没买 = 什么都没有买 (to not buy anything [in the past])
什么都不买 (to not buy anything [present or future])

Note that 没有 is an exception here, e.g. 明天什么都没有 (to not have anything tomorrow [future]) is okay.  I believe 什么都没是 is also incorrect (there's a mismatch between 没 and 是).
(Please note, I'm just a learner.  Hopefully if there's any misunderstandings on my part, someone can point them out to me in the comments.)