I have a LowePro bag I carry my gear in. This is plenty to protect it. I carry my camera everywhere, it doesn't leave my side. Customs, as has been stated, may ask you to turn it on and show them that it works and is in fact a camera, no big deal, hasn't happened to me in a long time.
If you have house insurance, you are covered. Remember, you have to deal with your deductible if anything happens. I have never covered my logos because my camera strap is a neon sign for "steal me", the rest doesn't matter. If you get a solid black strap, then consider covering the logos. I am not aware of thieves focusing on one brand, they typically are opportunists. Don't leave your bags unattended, ever.
The only time anyone has tried to steal my gear from me was at a World Cup celebration in a crowd where a moron tried to grab the strap and run. He missed. I had my family in tow or I would have stuck him good with my wood flag pole. Probably best he got away! LOL
Unless you are going to a high crime area, most tourists carry cameras and you don't hear too many bad stories. Leave your gold at home, don't flash your cash, wear a cheap watch and try to keep a low profile.
Oh, and ENJOY YOUR TRIP!