ℹ More Info.
More than ten years ago Apple included the iChat app fro the first time with the OS.
It was then just a text chatting app and Apple arranged with AIM/AOL that Apple IDs issued by Apple (@mac.com back then) could be used as valid AIM Screen Names.
Back then the app was exclusively AIM based or the Bonjour (then called Rendezvous) for LAN based Mac to Mac chats.
iChat 2 added Video and Audio Only chats.
iChat 3 added Jabber and GoogleTalk could be made to work with the app.
iChat 4 added the Google option to Add an account due to Google's strange Server name issue.
iChat 6 adds Yahoo (badly) as it only does text chats and no other features.
Mountain Lion (OS X 10.8) brings iMessages added to the Folder and a reworking of the Chat window (forcing tabbed chats effectively) and the new name of Messages. The App numbering continues with it being called version 7
The iMessages account tends to ask for an Apple ID as you open the app for the first time. (it will bring older accounts across from System Preferences > Internet Accounts if they exist).
The iMessages account in the Mac version and the Messages app (and it sending of iMessages) on the iPhone can be linked by adding the Apple ID to the iPhone after it's number has verified (Registered in fact).
Both Mavericks (OS X 10.9) and Yosemite (OS X 10.10) both list Messages as being version 8 thought the updates to the OS (which is not the version counting process used before this time.
The Yosemite version can do SMS, or Text Forwarding as it appears in the Settings of the iPhone Version, which, as has been said, has to be at iOS version 8 at least.
So.
Messages can do IM Chats over the AIM service, the Jabber service, the Yahoo service and the Bonjour connections to other Macs on your LAN as well as the iMessages service run by Apple. (This may require you to have different IDS for each service)
Re Question 2.
As BobHarris says it is supposed to tell you if you have Unread items.
Try Right Clicking the Dock icon and seeing if it lists the Chat it thinks you have not read.
If it does can the app bring it back to the Chat window if you select it ? (it may not list or it may not show the chat if it was fully deleted rather then dismissed)
If it does not go to Applications/Utilities and open the Activity Monitor app.
Find the Dock in the List and highlight it (it will be close to the Finder but not have an icon)
Once highlighted use the Quit icon top left (grey icon with what looks like a Stop Sign hexagonal on it.).
Confirm Force Quit.
The Dock will disappear from view on the Desktop and then reappear as the Listing appears in Activity Monitor as it starts up automatically again.
The Numbering on the Dock icon for Messages should be sorted now.
Re Question 3):-
I tend to be very pedantic about what is Messages (the app) and what is iMessages (the Account).
I can see that you mean that when iMessages is logged in but you have said the App in the way you phrased it and then could mean one of or many services in fact and will depend in some cases as to whether the iPhone is logged into them as well.
You might want to read this first Link your iPhone number and Apple ID to use iMessage and FaceTime - Apple Support
Enabling Messages on your iPhone will cause it to Register the Number at the iMessages server and it will read "Verifying" until this completes.
The Mac versions expects an Apple ID that is linked to a valid iCloud account (It does not have to be the same as your iCloud account).
Once this has Verified and logged in your can send to the iPhone Number from the Mac ID and vice versa.
Adding the Apple ID to the iPhone in Settings > Messages > Send and Receive then the Apple ID and iPhone Number become linked.
Pop ups will appear on the Mac version which your have to Accept to add the iPhone Number to the Mac's settings.
On both version there are then settings for which of those are to be used when Sending new Messages.
iMessages sent to either your iPhone Number or Apple ID will appear on both devices and can be replied from there.
The replies will show up (on the Contacts end) as coming from the ID it was Addressed to (independent of the "Start From" ID) - if someone sent to the iPhone Number and you sent back from your Mac it would still address your contact as coming from your iPhone Number even if the Mac version is set to "Start From" the Apple ID.
Of course you could also be logged into AIM/AOL with a Screen Name and/or a Jabber ID such as Google Talk and/or a Yahoo account and have either apps or a web page login on your iPhone


9:46 pm Sunday; January 3, 2016
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad