Is the "Remote Desktop" for screen sharing; is it included with the MacBook Air?

The Client is included in Mac OS X and is enabled in the System Preference>Sharing>Remote Management

The Administration app in the App Store needs to be paid for,

There is also VERY 'lite' version of this on every Mac - it has no Admin features. It is called Screen Sharing.app

It is hidden in Macintosh HD/System/Library/CoreServices/Applications/Screen Sharing.app

"Is the Apple Remote Desktop application for screen sharing?"

Yes. For screen sharing to work you would have to enable it and then it would only work within your own home or office network. For it to work over the internet would require a level of knowledge/expertise you won't have (no offence intended) regarding your firewall/router.

". . . for when Apple Support or one's local service provider asks to share the screen?"

I doubt if Apple Support would ever (maybe in very exceptional circumstances they might?) ask to control your computer remotely. As for your local service provider I'm guessing you mean your broadband provider? Again I doubt they would either as they'll tend to support their own product (ie: router/firewall) only assuming you're having connectivity problems/issues. Again in exceptional circumstances they might do but I'd bet they would use a different product altogether such as logmein etc. In my experience I've never come across any ISP support staff using a Mac in the course of their work. They'll use PCs as Windows and Internet Explorer are what most people use.

Richard Cartledge has answered your other questions regarding it but to keep it brief and simple, Remote Desktop is part of the OS and if there's an upgrade to it you should upgrade it. However it is not a critical part of the OS and if you chose not to upgrade it you can safely do so without causing any harm to your computer. Taking it further you can choose to skip upgrades for it whilst selecting other upgrades for your mac, again without causing any harm. You can also choose to upgrade it later on missing previous upgrades for it, again without causing any harm.

Ultimately what you chose to upgrade is entirely up to you.

". . . the broadband provider I'm using . . . are set to work with Windows only?"

Not necessarily. As already mentioned most people accessing the internet use PCs and Internet Explorer. Therefore it makes sense as a broadband provider to use your support resources to aid a user base that is mostly PCs. Occasionally you may get lucky and get a support technician who uses a Mac for personal use and/or knows a little UNIX/LINUX. In that situation they may be able to help? But that's not a common situation. In most cases they can't help because they're either not 'supposed to/allowed to' or it's outside their experience/pay levels. This is my experience and opinion only and you should not think that all broadband providers and their technical support staff are they same.

Apple on the other hand can only help you so far. They're not broadband providers and once they eliminate the OS, the reliability of the Wi-Fi/Ethernet connection to your home router they can't really do much else other than advising you to contact your provider. This should be your first port of call anyway. In the past I've supported Mac users experiencing internet connectivity issues that were all too often caused by simply not paying their phone bill. Some were non-plussed as to why their internet was not working even then.

The Remote Desktop upgrade is not large and it may take up slightly more or less space on your hard drive and/or remove/add to its bits and pieces as part of the upgrade. Apple do publish useful support articles as to what the upgrade does and contains in a fairly comprehensive way and these are freely accessible on their website. What you could do to find out what impact the upgrade has in terms of size, is to mark the difference in the hard drive capacity before and after the upgrade.