Can a country decide that they now starting to use USD for example, then enter USDs into their circulation? Is there any international law that stops that and to what extent (can use the currency, just its name etc)
Yes, they can. There are many countries that do not have their own national currency, as described for instance in this article.
There are basically two ways of adopting a non-national currency.
The Euro is used in a number of countries that together decide on currency policy, so individual countries have not given up total control over it.
There are other countries that simply adopt a foreign currency (often the US dollar) without any influence on financial policy. The advantage is that you benefit from the stability in value of the adopted currency without it being influence by your own economy, but of course the disadvantage is that you cannot influence the value of the currency.
As far as I know, there are no laws stopping such adoption of another currency.
Is there any international law that stops that and to what extent (can use the currency, just its name etc)
International law isn't really the problem here. Currency only has value in that people give it value. The home country of the currency won't accept currency issued elsewhere. This makes foreign-issued copies of the currency nothing more than counterfeits. In theory, the adopting country could give it value by accepting it there, but as a practical matter, countries with that kind of credibility don't use other countries' currencies.
Some countries adopt the currencies of other countries without trying to issue them. @oerkelens found an article on possible consequences of Scottish secession that lists countries without their own national currency. A fair amount of them are European countries using the Euro, either as part of the currency union or by trading in it.
Countries that don't issue their own currency or that peg their currency to another country's need a large supply of the real currency. They tend to get this by being large exporters relative to their own size. This is why most countries like this are rather small. There are exceptions. China pegged its currency to the dollar for many years.
Countries have significant ability to prevent others from issuing their currency. They can refuse to accept currency without provenance. But they can't restrict people from using their valid currency. That is after all what the currency is intended to do. Those with it trade to others in exchange for goods. If you disallow the exchange of currency, you disallow the exchange of goods as well.
This is quite commonly done, both in a hard way by out-and-out using the other currency, and in a soft way, by tying (or "pegging") the value of their currency to a specific value of another currency. Usually you'd want to use a big strong currency to do this with, which in today's world means either the US Dollar or the Euro.
Wikipedia has a nice list of such exchange schemes and the countries that use them. Currently there are 10 nations besides the US using the Dollar, and 6 using the Euro. Most of those extra Euro users are small non-EU countries in Europe. There are 32 countries (mostly in the Americas and Middle East) that have tied their currency to the Dollar, and 21 (mostly in Africa) that have tied theirs to the Euro. There are several more that have their currencies very loosely tied, the most notable of these is China which has a peg, but is known to move it around.
China actually shows the limits of this parasitical approach, in that when the parasite gets big enough, the host is liable to start having an issue with it. The tied Yuan has become an issue in US/China relations as the Chinese economy has grown. It hampers the ability of the US to implement monetary policy on its own currency. Current trajectory is for the Chinese economy to pass the US one in the next few years, at which point the tail could very well end up wagging the dog.