First of all, you should have specified "Google Cloud Messaging" (or C2DM in case) in the title of this thread, rather than "Android Push"...
1) If you don't set any custom icon, devices borrow, I don't know why, icons from other installed apps (the icon of Google Talk, sometimes). You can set your own icon from client with the following code:
String app_name = context.getString(R.string.app_name);
int icono = R.drawable.ic_stat_notify;
long time = System.currentTimeMillis();
Notification notification = new Notification(icono, app_name, time);
And you should follow the official Android Design Guideline to design that custom icon, usually designing three versions of the icon (one for devices running v2.2, another for devices running from v2.2 to v3.0 and another for devices running from v3.0).
2) Yes, that's exactly the collapse_key
purpose.