>When our hyperkähler 4-fold is no longer compact, is its
>instantons moduli space (maybe with some constraint about
>decaying condition) still hyperkähler?

The "decay constraint" condition is called "gravitational instanton".
A gravitational instanton is a complete hyperkähler metric on
a 4-manifold with $L^2$-integrable curvature. There is a lot
of literature on gravitational instantons and on instanton
bundles on gravitational instantons. The gravitational instantons
are classified according to the asymptotic growth of a volume
of a ball: when it grows as a 4-th degree (like on Euclidean space),
it's ALE instanton, degree 3 is ALF, and the classification becomes
very detailed and intricate for smaller degrees. ALF and ALE spaces
can be constucted as hyperkähler quotients of flat hyperkähler
spaces, for lower degrees it's probably false.

The space of instantons on ALE space is hyperkähler:

P.B. Kronheimer, H. Nakajima, [Yang–Mills instantons
and ALE gravitational instantons](https://doi.org/10.1007/BF01444534). Math. Ann. 288 (1990),
263–307.

The same is true for instantons on Taub-NUT space, 
which is ALF:

Cherkis, Sergey A.
[Moduli spaces of instantons on the Taub-NUT space](https://doi.org/10.1007/s00220-009-0863-8). 
Comm. Math. Phys. 290 (2009), no. 2, 719–736.

and on arbitrary ALF space:

Cherkis, Sergey A. 
[Instantons on gravitons](https://doi.org/10.1007/s00220-011-1293-y).
Comm. Math. Phys. 306 (2011), no. 2, 449–483.

There is no framing in these results, you need to consider
instantons with $L^2$-integrable curvature.

I am pretty much sure one can prove the same result
for all gravitational instantons in place of ALE and ALF,
and in fact I have a proof (which is pretty easy), but
I am sure it's already published somewhere. The literature
is a mess.

>For some noncompact kähler surface that are not
>hyperkähler, for example blowup of $\mathbb C^2$ at origin,
>does its framed instantons moduli space still admit a
>hyperkähler structure?

Mostly not, in fact, it can easily be odd-dimensional.
One could imagine results like that (for instance, $CP^2$
is not hyperkähler, but instantons on $CP^2$ are
hyperkähler, if the framing is chosen right), but 
this would probably mean that the complement 
to the curve where you have chosen framing 
is hyperkähler.