Bartley77:Do the matches played in the play now area Monico, Paris, Casablanca rooms ect. Count as ranked rounds?
Nope.
Matches aren't ranked rounds but "rated" at most, and all the special mobile games, "Rooms" and "Head to Head" are deemed practice only, meaning that they do not count in any list AFAIK.
Ranked rounds must be strokeplay, 9 or 18 holes. Tournament play is included there as well as MP strokeplay but no practice rounds. Furthermore, Par3, Par5 and CC Custom Courses may be played as ranked rounds but do not count towards the average.
Bartley77: How can you find the number of rounds you’ve played in your current tier?
You'd have to count them. On PC, it's easy in the Score History (a tab on the profile page), and you may copy&paste the list to a spreadsheet or a document if you need to save more than 200 games.
On mobile, ?screenshot?
Or, note the number of RRs in your stats when you start a new tier and observe the progress. Mind you that Par3, Par5 and CC Custom are included in that number.
Dunno why you are asking, but if you are keen to know when you will reach saturation to be promoted, don't hang in there too much. Until Master, it may go almost by itself, Tour Masters may have more struggle with the scores than with the number, and for Ls and TLs (saturation at #500) it may be a long grind where fun is a motivator while counting the grind is not.
Hit it straight!