Many times, it happens that you have got 6-7k views on your question, and need a few more to get a golden badge. Now, if it is unlikely that the question would get that more views on its own, am I allowed to explicitly advertise it?
Well ask yourself an honest question, really - are you here to play or are you here to learn/contribute?
Nobody is stopping you from advertising it on any other social networks, feel absolutely free to do so. If you advertise it here just to get more views in order to obtain a badge and NOT to get an answer, you may get some negative feedback.
But bear in mind, that it's your time invested in an activity that this site hasn't been really designed for. The award system motivates people to ask better questions and respond with better answers. It wasn't primarily designed to be a social game per se.
am I allowed to explicitly advertise it?
Allowed? Sure, it's your question. While the question also belongs to the community as whole (so you're not allowed to just delete or alter it without a good reason), nothing is stopping you from posting links to the question anywhere you want.
Is this good idea? If you ask me the answer is "no". Paying real money and/or spending real time on getting a virtual badge isn't a good practice in my opinion. Badges should come as a pleasant surprise, as Jeff once said. While hunting for badges is also good sometimes, spending too much time and efforts over this might reduce your overall "efficiency" towards SO/SE as whole.
This depends entirely on the website you promote it on. We can't answer this for other sites, you'll have to check yourself what the rules are for each site.
In case you want to promote your own question on Stack Exchange, this is problematic in most cases. Bounties are the supported method to gather more attention to your question, almost all other methods are inappropriate. What you shouldn't do is e.g. commenting on unrelated questions or promoting the question in chat rooms you aren't active on.
Unless you post spam suggesting to take a look at your questions, or be annoying (say, in the chat) posting continuously posting messages regarding one question just to gain more views, that's absolutely allowed to advertise your questions on, maybe, sites of your own, and even encouraged.
There are some badges regarding sharing your questions - Announcer, Booster, Publicist (which, by the way, won't work if you spread the links within SE). Also, every post has "share" under it that allows you to either copy the link or share it via social networks in convenient way. You can put those links on your site and have no problems.