Let's suppose that I have an APK file. I decompiled it but I have very obfuscated code. Is it possible to analize this APK in such way that application will be executed step by step (instruction after instruction with breakpoints between each instructions) and the debugger will show me equivalent of compiled instruction in my obfuscated source code that I will able to deobfuscate it (give methods correct names for example) on my own?
Yes, this is indeed possible.
IDA Pro, starting from version 6.6 supports source code level debugging for Dalvik Bytecode.
You can find more details here: https://www.hex-rays.com/products/ida/support/tutorials/debugging_dalvik.pdf
Also, you could use dex2jar to get the Java code from the APK. Then you can manually deobfuscate the code by writing short Java Programs which will decode/decrypt/deobfuscate sections of the code.