Your IIS ASP.NET application is actually not a legacy app. If it's ASP.NET you can simply host it in an Azure web role. Download the SDK, create a new cloud project and choose web role. This will be able to host your ASP.NET site.
For this you'll need to "rent" a hosted service with N instances of a web role:
Yes you can manage your server through remote desktop. But you wouldn't want to do that, seen that the Fabric Controller (the software managing Azure) might remove the machine you made your changes on for a clean installation. Your alternatives in this case
are startup tasks or the VM Role:
You need to modify your solution - you need to create cloud project and link it with your existing project.
And yes, you can host any kind of software. Windows Azure has a "bridge" between PaaS and IaaS called Virtual Machine Role. You should prepare your VHD-image with WS2008R2 and your application inside and then upload it to cloud. Recently was announced that
VM Role will support Linux as host system, so and even this scenario is ok.