Run Windows program on Linux in a secure way

The host has complete control over the guest. So if there is malware running on the host, the guest is compromised as well.

In practice, most malware out there only tries to infect common software, such as the OS itself, web browsers, media players, etc. Virtual machines are such a rare case that malware authors typically don't bother, especially given that the diversity of systems that could be running inside the VM means a lot of work for the malware author.

However, this may change as virtual machines become more common. For example copy-paste would make a relatively simple infection vector (you think you're copying some innocent text into the VM but the malware makes you in fact paste a command that opens the gates; also, consider that by spying the host clipboard, the malware has access to anything you copy-paste even from within the VM to within the VM). And if you're the victim of a targeted attack rather than some common malware, all bets are off.

If you have a network link between the host and the guest, the malware may attempt propagate that way, like it would to any other machine in your local network.

Yes you can use Virtual Box images portability across different host OSs. It's a little bit complicated because you need more than the hard drive image, you also need the virtual machine configuration file.

• If this is a one time transfer you might consider the export / import options to help you get all this stuff transferred.
• If this is a transfer you hope to make many times, you might look into VirtualBox's teleport function that allows you to transfer running virtual machines back and forth between different hosts.

  Starting with version 3.1, VirtualBox supports "teleporting" -- that is, moving a virtual machine over a network from one VirtualBox host to another, while the virtual machine is running. This works regardless of the host operating system that is running on the hosts: you can teleport virtual machines between Solaris and Mac hosts, for example.
  
  Teleporting requires that a machine be currently running on one host, which is then called the "source". The host to which the virtual machine will be teleported will then be called the "target"; the machine on the target is then configured to wait for the source to contact the target. The machine's running state will then be transferred from the source to the target with minimal downtime.