The moderation chat room was initially created to facilitate discussions regarding moderation and moderators when we used to have lots of repeated meta posts on it.
As known to you also from the very first day of the chat-room:
That a time our main chat-room used to be active with general discussion and hence this room was created to handle the traffic and try to solve disputes. As the time passed eventually the room become inactive and freezes which was unfrozen sometimes as per requirement.
Until 12 May 2018. This room used to be private gallery (only allowed users can talk). On 12 May 2018, this room was made public chat-room as per the discussion Have a chat room where all the site moderators are contactable. It used to freeze sometimes due to inactivity when there was no moderation discussions and disputes and was unfrozen by moderators when required.
In August 2019 it made frozen due to some non-constructive or confusing discussion going on and recently unfrozen as mentioned by Glorfindel in his answer.
A few days ago a mod left a message for me in this room but there's no way to respond.
Yes, you can now respond to the messages. Or anytime ping us on main chat-room in case if room found frozen.
Going forward, should issues with moderation (declined flags, post closures, etc.) be discussed in this chat room or can they be posted on the meta site as well?
You're free to discuss on chat as well as on meta. Pinging in chat would be useful for a quick understanding or clarification of small issues while for major issues or issues concerning community, policy etc. asking on Meta is recommended especially when something needs to be documented.
There were a few occasions when a message posted in this chat room was ignored by the mods so a user may eventually end up posting about their problem on meta. So should users first try to discuss/resolve their issue in this chat room and post on meta if there was no resolution?
You're free to post your queries on meta (make sure to search for similar already discussed problems which may answer your queries). There is no such rule that you should first try to resolve in chat but doing so (real-time discussion) would work sometimes depending upon what you're looking for and availability of mods on chat-room.