@sanket good questions. You are hitting some limitations and quirkiness of schedules and thermostats. I''ll explain in more detail below.

1. Unfortunately you can't currently clone schedules in the OpenStudio GUI. We do plan to fix that, but as a result if you want one schedule that is a slight variation on another, it is a bit more work. We do have an issue open for this. Making a new schedule is also a bit confusing. The green "+" makes a new schedule, but then it is important that you choose the proper "Class" and "Schedule Type" based on how you want to use the schedule. if you don't do this correctly the GUI may not let you apply the schecule, or if it does, energy plus may fail. See the image caption for more details.
2. This is another peculiar behavior what we need to address. In the SketchUp plugin our templates are pre-loaded with a few thermostats that are treated like a shared resource. if you apply a thermostat to 5 thermal zones, they all use the same thermostat. When you open this model in the stand alone OpenStudio application and change one of the schedules, you are changing the shared thermostat, so it will change in all zones. If you turn off a thermostat or delete the setpoint schedules, it will be disabled in all zones. There isn't any indication of this since the thermostat object itself is kind of hidden, with the user only seeing the heating and cooling setpoint schedules. If you start fresh in the OpenStudio application and assign thermostats here, behind the scenes we are making a unique thermostat just for that zone. Many zones may share heating and cooling setpoint schedules, but the thermostats are unique. As a result anything you do in one zone wont' impact another. We also have an issue open for this. We do also have a measure that lets you batch assign thermostats to all zones in a model in the OpenStudio application. It doesn't use a shared thermostat, so it is as if you added them by hand in the Openstudio application, so you can then alter specific zones.

This is the dialog you see when you make a new schedule. choosing the "Class" and "Schedule Type" will set the type limits for the schedule and determine what type of objects the schedule is intended to be used for. Currently you can't easily see or edit this information after you make the schedule, although units will be displayed when viewing a profile and will be affected by the si/ip units preference.

For reference, here is what the set thermostat measure looks like. It can either apply to a single zone, or to all zones.