What are very short programs with unknown halting status?
Yes. ​ This page says there are 98 5-state Turing machines whose halting statuses are unknown. ​ Annoyingly, it does not give any examples of such machines, but this 26-year-old page gives 2 5-state Turing machines whose halting statuses were apparently unknown at that time. ​ (Searching for "simple counter" will take you right between those 2.) ​ I copied them here in case that link goes down:

Input Bit   Transition on State     Steps            Comment
             A   B   C   D   E

    0       B1L C1R D0R A1L H1L   &gt; 2*(10^9)       ``chaotic''
    1       B1R E0L A0L D0R C0L

    0       B1L A0R C0R E1L B0L       ?        complex ``counter''
    1       A1R C0L D1L A0R H1L