The problem was the animation iframe on your home page. It is apparently mangling the FTH system cookies, which is why you are getting logged out. It really has nothing to do with Membership Management... you would have gotten a similar error messages in other modules eventually, also.
I was able to log in as the Admin from your home page, just do a refresh, and then I discovered I was logged out. This is a likely sign of a problem in the home page content, which turned out to be the Animoto iframe. I removed it, and everything works fine now... no log outs.
Whenever you use third party widgets and content, you have to take care to test them with FTH to make sure that everything will be compatible. We are not in the position to do that testing, and even if we were, there are enough widgets on the internet that we would not likely want to be testing them all anyway.
In any case, as I stated above, I removed the iframe html... it is below for your reference and if you want to fiddle with it to try and get it to work without mangling the system cookies. From my review of its online docs, it requires the use of Adobe Flash, which is generally speaking a bad idea, anyway. I would recommend that you find an alternative widget.
Code:
<table border="0" cellpadding="0" style="width:100.0%">
<tbody>
<tr>
<td>
<p style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="720" scrolling="no" src="https://animoto.com/play/Ak2ch1Fdrnx9Az8xqL3pNg" width="640"></iframe></p>
<p style="text-align:center"><br />
</p>
</td>
</tr>
</tbody>
</table>