Setting DISPLAY in systemd service file

An application needs two things to open a window on an X display. It needs to know the location of the X display; that's conveyed by the DISPLAY environment variable. It also needs to authenticate with the X server. This is conveyed through a cookie, which is a secret value generated by the X server when it starts and stored in a file that only the user who started the X server can access. The default cookie file is ~/.Xauthority.

If your X server is using the default cookie file location, then adding Environment=XAUTHORITY=/home/dogs/.Xauthority will work (assuming /home/dogs is the home directory of the user who is logged in under X). If you need to find the location, see Can I launch a graphical program on another user's desktop as root? and Open a window on a remote X display (why “Cannot open display”)?

Alternatively, running the program as the user who is running the X server will work, provided that the cookie file is in the default location (if not, you'll have to locate the cookie file, like in the root case). Add the User directive (e.g. User=dogs).

Of course the service won't run if there isn't an X display by that number owned by the user you specify.

It's rather bizarre to start a GUI program from Systemd. It wasn't designed for this. GUI programs live in an X session, started by a user. Systemd is for system processes. You should experiment with daemons instead.