Whether google play services is a built-In installation in every android phone?

Question 1

No, the Google Play Services are not mandatory. There are some devices out there that don't offer them as they cost licensing fees and you need to setup the device so it "promotes" Google services. In general Android is open source and free and can be used without these services. But as they are tremendously useful most people want to use them.

Android devices without them are e.g. the Amazon Fire tablets, a lot of local Chinese devices, the new Nokia Android phone (who will ever buy one of these and why??), or most devices with custom ROMs (e.g. Cyanogenmod). As the services add a lot to Android and in fact are seen as substantial to most users, adding them afterwards is possible while not always legal (but why would Google prosecute people who want to use their services?). It is not legal to deliver these services with custom ROMs but you can find them on certain download sites. It is not always fun to get the exactly correct version and to get them working.

Question 2

Usually all android phone will have play service as built in app.if not You can use the below app to use GCM,Map etc. in your app

https://play.google.com/store/apps/details?id=com.google.android.gms

For more details

http://developer.android.com/google/play-services/index.html

Question 3

Google Play Service isn't mandatory for Android to run, I think it's 98% included in any Android OS because you need it for others Google Apps (Maps, Play, Google + etc)

But as you can see, it's a standalone app : https://play.google.com/store/apps/details?id=com.google.android.gms&hl=fr

So if you use it, you can check if it exists on the mobile before doing anything (and perhaps popup a Dialog to invite the user to install it)