All three cases are actually different.
In the standard (Tokyo) dialect,
死なせる is the correct form, and
死なす is marginal. If
死なす is allowed in any case, then
死なせる will not be allowed, and hence these should be considered different dialects/idiolects used with the same meaning. It is the causative form of
死ぬ (See below).
In the standard dialect,
生かす is the correct form, and
生かせる is completely ungrammatical (unless you are talking about the potential form).
Here, you see the causative morpheme
-(s)ase-. When the verb stem ends with a consonant, the form
-ase- is used.
kak-ase-ru 'cause someone to write something'
kag-ase-ru 'cause someone to smell something'
kas-ase-ru 'cause someone to lend something to someone'
kat-ase-ru 'cause someone to win'
kam-ase-ru 'cause someone to bite something'
sin-ase-ru 'cause someone to die'
nom-ase-ru 'cause someone to drink'
When the verb stem ends with a vowel (
e), the form
-sase- is used.
mi-sase-ru 'cause someone to see'
tabe-sase-ru 'cause someone to eat something'