iPad surisingly locked to AT&T service

ATT does not lock iPads, especially if it has already been working with your carrier. There must be some other explanation for your issue. Where exactly did you get the iPad? Perhaps it’s not locked but blacklisted.

I’ve bought it from the store in official package. 

Have receipt (somewhere)

At least can anybody post an email address for human support (or chart link)?

Sorry, ATT does not do customer support via email. You can click the contact us link at the bottom of the page to get to chat.

I would suggest you check the IMEI to see if it has been listed on the universal Blacklist.

Hi! I've checked IMEI. Status unknown 

You can click the contact us link at the bottom of the page to get to chat.

I've tried it, but get only AI there. Do you know "magic word" which calls human in chat?

As a non customer, outside the USA, you cannot get live chat.    AT&T doesn't put any effort for non customers.  Try social media, Facebook or Twitter.   Otherwise, it's wifi only from now on 

SMM specialists from Facebook pointed me to chat with AI support only and to the phone line. Stalemate :)

As for blacklisting: is there any online service to check was it blacklisted or not?