How do I login locally to a Mac that is tied to a Windows Network Domain

Was the account set up as a network account or mobile account. Network account is the Windows equivalent of Roaming Profiles in which everything is set up on network shares, and doesn't really allow for "disconnected" logins. Mobile accounts are the equivalent to the local profiles for Windows network users, in that login credentials and resources are "cached" locally, and will allow a disconnected login. If you have a network account, you can convert a network account to a mobile account in the Users and Groups settings of System Settings. I would provide a screen shoot, but I don't have an example to show. I have done so in the past when users did not create a mobile account upon first login/creation of account. You will need to be connected to the network to do this.

There is some discussion of the differences between the two here - OD: network account vs. mobile account

Not sure about the Home icons, but this is how you convert: (OS X Yosemite: Create and configure mobile accounts)

A mobile account lets you access your server based network user account while remote from the network and creates a local copy of your network home folder. Whether or not you’re connected to the network, you use the same user network name and password when you log in, and the desktop and other items on your Mac look the same.

Before you start, ask your network account server administrator to set up a mobile user account for you.

1. Log in to your Mac using your network user account.
2. Choose Apple menu > System Preferences, then click Users & Groups.
3. Click the lock icon to unlock it, then enter an administrator name and password.
4. Click the Create button next to “Mobile account,” then specify which folders to sync and when.To change settings for syncing folders, repeat the steps, then click the Settings button next to Mobile Account. If the Settings button isn’t available, the network account server doesn’t allow changes to the mobile account settings. Ask the network account server administrator for help.

Suspect you have a permissions issue between AD and the home directory. IT has been over a year since I last had this issue crop up, and do not remember my exact steps I took, but did you convert, then reboot and log in with he "mobile" account, so it could create the local folders for the suer?

K...not knowing your setup, and not wanting to get into a lot of details, your best bet may be to just delete the existing account out of Users and Groups, make sure your AD settings are set up this way (may require you to unbind, set the settings, then bind again) (make sure "Create mobile account at login is checked and confirmation)

and then recreate your mobile account again by logging in with the AD account after a couple of reboots. Make sure when you login for the first time you select the Create Mobile Account option when it pops up. In my experience, first time you login you have to use the <Domain>/<UserName> format, then every other login can just use the <UserName> format, since it caches everything and just verifies credentials.