"Happy Birthday" = "С Днем рождения!"

As for "The Cake is a Lie", I googled out the following:
The Cake is a Lie is a catchphrase popularized by the game Portal, and is often used to convey the message that a promised gift is being used to motivate without any intent of delivering.

I was unable to recall any equivalent Russian idiom. There is a slang/colloquial word "динамщик/динамщица" for the person making false promises.

I have no idea how it relates to birthday though.

Edit: there is a limited idiom "морковка на палке":
86567904_Carrot.jpg
This device presumably makes donkey moving forth.