Still no refund, and when checking status on IRS website it writes: We apologize, but your return processing has been delayed beyond the normal timeframe. What to do?

Only the IRS can give you any information about your return. They are telling you that you will have to be patient. 

thanks. But calling IRS [phone number removed] sends you to a labyrinth of messages you can not escape, and which is completely not helpful for this case.

The website also does not show anything. And there is no chat or email option.

Waiting from April to end of September, and nothing yet.

How long should one wait? and is there an option to learn what is happening?

IRS has changed the prompts on the main number.

New procedure --
While robot is giving you options, interrupt with "I need to speak to an agent"
When it asks for your reason,
Say "Account Management".

You will get another long spiel, and then the option to hold or be called back when agent is ready.

@ganar 

just tried, alas, they closed this door;

it again sends you to that loop of unrelated options.

Is there some legit way to contact them? How can an organization take our money without any possibility to contact them in case of their problems.

You may want to contact a Taxpayer Advocate in your area.  See this IRS website for Taxpayer Advocate - https://www.taxpayeradvocate.irs.gov/

"Say "Account Management"."

Instead of "Account Managment", try " I have a question about Identity Theft".

@ganar 

P.S. If the wait time is too short, you may not  get the "call-me-back" option.