How about this: My dreams are getting true. I think it is correct.

To me, the best option is "My dreams are coming true". You can also say, "My dreams are becoming real" however you cannot use "My dreams are starting to be alive" as this isn't correct English. Another option is "My dreams are becoming reality".

This is a commonly used term and as a native speaker in the US, I've only ever heard "My dreams are coming true". There are fancy, more eloquent ways of communicating the same idea, but you'd only find those in novels and older books. "My dreams are coming true" is the only colloquially correct one you mentioned above.

Можно рассмотреть и такие варианты: Dream that is becoming reality. My dreams are ever more real - Мои мечты всё больше становятся рельностью My dreams become true - Мои мечты становятся реальностью I make my dreams come true - Я воплощаю свои мечты в реальность