The MySQL error suggests you are trying to install OpenSEF again.
That is very odd if you are simply trying to make some menu changes.

Are you sure your database backup is OK?
I took a look at your site (very nice BTW).
I see you have two languages and I am guessing you are using MambleFish.
I do not know of anyone who has MambleFish working with OpenSEF.
We have not tested it and probably will not test it until JoomFish is released.
At this point I suggest you set OpenSEF Enabled? to Off.
I also see you are using WordPress for your blog - with fancy URLs.
I am sure this requires an htaccess file from WordPress.
Using standard Joomla core SEF (and OpenSEF) requires using another htaccess file.
These htaccess files can interact with each other and cause problems.
I know the JD-WordPress bridge component does work with standard Joomla core SEF.
It is used on the Joomla Developers site for the developer blogs.
It also probably works with OpenSEF On as the JD-WordPress developer is also the OpenSEF Lead Developer.
This does not mean you would have fancy URLs, just they will not conflict.
I have not used JD-WordPress so there may be other issues.
i.e. Don't know if Permalinks work, fancy URLs, etc.
But no Joomla errors.
I assume you have some htaccess rewrite errors as the Home link from your blog pages results in a 404 error.
Some of these rewrite issues may be affecting your Joomla Administrator.
Standard Joomla core SEF has no effect on the Joomla Administrator links.
I am wondering if something you have in an htaccess file
is affecting the J Administrator.
What would I do?
Turn everything Off and start adding things back one-by-one.
Turn Off WordPress fancy URLs and Permalinks.
Turn Off OpenSEF.
Turn Off standard Joomla core SEF.
No htaccess files anywhere.
Does the site work with no SEF On anywhere?
If yes, at least you have found the source of the error.
If no, it is another issue.