They verbatim say they want to work behind the scenes in the company to change the culture. That's not advising or just "having influence". It's simple...they do not have a right to do this as investors unless it was part of their terms.
That's ridiculously. They certainly have a right to do that. Sure, it depends on management etc.'s willingness to listen and take the advice, but they absolutely have the right.
No they do not have that right. They have the right to if management is willing as you say, which they aren't, in this case. They do also have the right to play their part in replacing the board, and they do have the right to try and pursue board membership. But they aren't entitled to work within the company to change its culture simply because they're an investor. Companies could not accept investment if this was a right extended to investors.