Does the mini box support HDMI resolution levels, i.e. 1920 x 1080 60Hz?

HDMI is kind of an odd place to start.  The Mini Box is HD capable.  HD is any image resolution over 480 vertical pixels with Interlaced scanning (480i).  480p is considered Enhanced-Definition Television (EDTV).

HDMI is an odd place to start because it's a digital cable capable of much more than the HD standard.  For example, HDMI v2.1 can support UW10K resolution (10240 × 4320).

All versions of HDMI are backwards compatible.  That doesn't mean you'll get an UHD format from your mini box just because you connected it to an UHD TV.  The 2 devices will negotiate the best possible format (HD).

As far as your monitor not detecting the HD signal, you may have a setting overriding any auto-negotiation.