It does have an analog modem (should be mentioned in the manual but it's also listed on the feature list on the website). Just run a genuine test with the plugin. You can as well check the image via menu > information > about. As soon as you see SIM in the name of the image it is for sure a fake.

for your other problem I recommend a bootlog. If there's a problem it's usually visible in the log.

If the box went dead after 5 minutes of use, you should contact the dealer! At least in Europe the dealer has to repair or replace the device if it fails in the first 6 months!
Alternatively you may contact DMM via their ticket system: http://www.dream-multimedia-tv.de/en/support They can also provide you with information whether your box is genuine or not (based on information on the dealer, serial number, production date, etc...).

You might also upload some pictures of out- and insides here. Maybe someone can compare it to an original then.

In generall production in Taiwan is correct. I think every dreambox except the 8000 were/are produced in Taiwan. About the modem I am not sure. I've read about boxes without analog modem. But this might only be true for the 7020HD or the 820HD and 7080HD. I can't find any current information about that anymore