Are the users who get these notifications random?
No, it's not random; just like on the main site, flags are visible to moderators and to 10k users.
This is based on local privileges on the chat server. On Stack Overflow and Meta Stack Exchange chat, only local reputation and moderator privileges are considered. On Stack Exchange chat, combined chat reputation and network moderator privileges (moderator on any site in the network) are considered.
How many valid/invalid votes does it need before a user is suspended or a flag is cleared with no action?
For the first part, see How many chat flags are required to remove a message/ban a user?. Basically, a net six flags are required (each "valid" mark is considered a flag and adds one; each "invalid" mark subtracts one), and a moderator's flag will be instant and binding.
For the second part: all flags on a message are cleared when the net flag count (flags minus "invalid" marks, as defined in the other question) goes to zero, with the exception that at least two "invalid" marks are required (i.e. one flag + one "invalid" mark doesn't clear). A moderator's "invalid" mark will be considered binding regardless of the above exception, and will dismiss all flags.
Are there any consequences to flagging a vote and having your flag declared invalid, like the main sites' net helpful flag count?
No, currently there isn't such a thing. We're keeping this in the back of our heads, so at some point there may be, but currently there is not.
How do automatic suspensions work? How long are they?
If a user's message reaches the flag threshold and gets removed, the user will be issued a 30-minute automatic suspension. This suspension applies to all chat rooms on the same server. The suspension will be clearly identified as "automatic", and the suspension message will link to the specific message that was removed for the user only (not shown to others).
If a second or additional message is flagged and deleted within 24 hours of a previous automatic suspension, the user will be issued a 1-hour automatic suspension, and moderators will be notified. Automatic suspensions do not extend beyond one hour, though many automatic suspensions in a row may prompt a moderator to issue a manual chat suspension. Moderators also have the ability to lift automatic suspensions if necessary.