Remote desktop without a password

"Yes, this is possible.

By default, Windows will not allow the logon over a network with a blank password. There is a KB article that details how to allow blank passwords for network logons.

You can disable blank password restrictions by using a policy. To locate and change this policy:

Click Start, point to Run, type gpedit.msc, and then click OK to start the Group Policy Editor.
Open Computer Configuration\Windows Settings\Security Settings\Local Policies\Security Options\Accounts: Limit local account use of blank passwords to console logon only.
Double-click Limit local account use of blank passwords to consol logon only.
Click Disabled, and then click OK.
Quit Group Policy Editor.

NOTE: By default, this policy is on (enabled).

Under the cover, in Registry, this is controlled by

[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Lsa]
""LimitBlankPasswordUse""=dword:00000000

Therefore, alternatively, this can be achieved by directly setting LimitBlankPasswordUse to 0. Setting it to 1 will restore the default behavior.

After you have enabled this, you will be allowed to log on using a blank password."