Just checked and the DMG file is sitting in the applications folder.
Sounds like you opened the dmg file and double clicked the app inside when you first launched it.
This does NOT install the app. I know newcomers do that because they think it's an installer like an .exe in windows, but it's not.
The dmg is a disk image file that mounts a virtual drive with the file inside. Typically, the file inside is the actual app file itself, all self contained. To install, you just drag that app file somewhere - usually people put them in the applications folder.
And you just run them from there.
What it sounds like you have done is run it from the dmg file. When you do this (and subsequently pin it to the dock) it is actually running from the virtual drive every time. This is why the icon disappears when you reboot. The virtual drive is I mounted and the app essentially doesn't exist, so there's no icon. When you try to run it it actually then mounts the drive and runs the app again from the dmg.
If this is what you have done, you will see the dmg mount the virtual drive on your desktop or finder sidebar every time you run it.
Open the dmg and put the app into your applications folder and run it from there.
Hope this helps.
Note: this is the typical mac installation style. I'm not sure of that's how chrome has done it. Some developers actually DO put an installer in their dmg (I.e Dropbox).