I have an idea of a genetically modified animal whose artificial features will solve certain practical problem. The combination of features is far from trivial. Can I patent (in UK, EU, US and the rest of the world) such an animal before investing into practical research and getting a required DNA changes? If so, can you point me to similar patents?
So your question actually raises two issues, which I will cover separately.
Can I patent a genetically modified animal?
In the US, it has long been practice to allow claims to non-human animals. A clear statement of this is at MPEP § 2105:
[From 1987,] the Patent and Trademark Office would now consider nonnaturally occurring, nonhuman multicellular living organisms, including animals, to be patentable subject matter within the scope of 35 U.S.C. 101.
There has been some recent development in genes. Naturally occurring genes are not patentable (per the 2013 US Supreme Court decision Association for Molecular Pathology v. Myriad Genetics). However, artifically created ones still presumably are.
An example of such a patented invention is US 8071839.
In Europe, animals are in principle patentable.
EPC art 53 provides:
European patents shall not be granted in respect of: … (b) plant or animal varieties or essentially biological processes for the production of plants or animals.
But, EPC r 27 provides:
Biotechnological inventions shall also be patentable if they concern: … (b) plants or animals if the technical feasibility of the invention is not confined to a particular plant or animal variety …
So an animal (or plant) which has some invention associated with it, but is not confined to a specific variety, is patentable.
An example of this is claim 12 of EP0264166, which claims "a mammal".
Can I patent an idea before working out how to obtain it?
The description has to contain enough detail such that a person who reads it could put the claimed invention into practice (in the US, 35 USC § 112; in Europe, EPC art 83). It is generally not enough to simply describe a solution without specifying how someone could get there.
In your case, the detail seems to be the precise genetic changes. Without this information, it is very likely that anything you claim would be insufficient, so any patent application would not be granted.