DC power adapter leakage safety
1 - I would say yes, it is probably safe, current kills, not voltage. These types of spurious high voltages are almost always detectable in ungrounded SMPS designs where capacitors are used which bridge between the HV and DC output stages. For quite a good explanation see the thread linked in the comments by Andy AKA.

As already mentioned, as good (and potentially life-saving, in other situations) advice - always perform any installations while disconnected from power. If required, offer the light up first and mark installation position / holes, etc.(while lit, so long as we are only talking safe voltages like here). Disconnect it, THEN install and test.

2 - See the link Andy AKA gave for more detail, but in short, you may be able to reduce / remove this with grounding resistance if available / where potential for touching of the electronics is possible. Alternatively, protect your strip with a case (again if possible).

You say the LED strip stopped working when installed? In such a case many strips are metallic backed. You may be finding that the strip is shorting to the window or by means of whatever you have used to install it (screws etc?). 

By no means a comprehensive answer but hope this helps :)