Can we install an unpublished iOS app on a remote device.?

Yes, you can. To do this you will need the UDID of all the devices where the App will be installed, and then you need to generate an Ad Hoc provisiong profile for those devices.

https://developer.apple.com/library/ios/documentation/IDEs/Conceptual/AppDistributionGuide/TestingYouriOSApp/TestingYouriOSApp.html

To send the app to your client, go to XCode > Product (Menu) > Archive.

Then open Organizer > Archives (Tab) > Distribute. Follow on-screen instructions. This will generate an .ipa file (executable) which you can then send to your testers (through email, or whatever other mean). He will have to install this .ipa by dragging and dropping it onto iTunes.

For further discussions look at this question.

Yes, you can.

1)Register client's device into device list in your developer account on apples site

2)Generate Ad-Hoc provision profile and send to your client with *.ipa file

3)Test it

Also take a look at 3-rd party services like TestFlight - http://testflightapp.com .Very popular and convenient way to work with beta-testers all around the world.

Answer is yes. You need to get UDID and add it to your developer portal as testing device. Download the provisioning profile and rebuild the app with revised provisioning profile. This app can be send to client for testing.