It depends which law applies
A contract is interpreted under someone’s law. For a competent internet company the contract will usually specify which law applies, if not, the court must decide where the contract was entered into - this may be where you are or where they are.
While clauses allowing unilateral changes are legal, the power must ...
Many statements made in commerce might be considered to be "false advertising", but not all will be found to be legally false advertising. False advertising is a statutory creation, so you would have to look at the laws against false advertising, in your jurisdiction or in the advertiser's jurisdiction. In the US, there are federal laws that govern this ...
...and according to their terms and services, the company has the
right to change their conditions of the services they offer.
You signed a legally binding contract when you first bought hosting services that included the "lifetime domain"service; and part of that legally binding contract was the clause that says the company can change their terms and ...
This actually happens all the time and isn't considered extortion.
Essentially, this is nothing more or less than a free service getting monetized after a while. This happens many times per year. For example, earlier this year the Have I been Pwned API has switched from free to paid because of the large amount of abuse: https://www.troyhunt.com/...
Extortion is a threat to do something unlawful
While threats to do something that's harmful for you are a key component of extortion, those threats generally (there are additional, separate provisions for threatening to exposing secrets etc) need to be unlawful .
For example, if I haven't paid the bills for some service and they send a letter "pay up or we'...
This is perfectly fine, unless you have a contract which states that they must continue to provide free service until a certain date.
If anything, they're doing the right thing by notifying you that things will change so that you can make other arrangements if needed. If they started charging you $20/mo. without your agreement, that would be illegal.
Is that extortion? false advertising? or in any way illegal?
Not at all. The owner of the site is simply exercising his right as outlined in the terms and conditions from when the user signed up. And giving users an option for continued use of the site (that is, for him not to exercise a right of which they were always aware) does not constitute extortion.