I read (and reply) to this forum on my computer and also my iPhone 5, but on my phone using Safari I'll notice trouble with picking up a thread at the last read message, even in the same mobile session. Any hints?

I'm not sure I'm following your steps, but in general Chrome works better with this site.

I'd load Chrome, but I've got limited memory on this phone. Basically sometimes I go back to read a thread and I'll be rereading the same messages until I get to the new messages and this is after reloading the unread posts page.

Maybe the browser is trying to save power / bandwidth by aggressively caching? The last read position is store in the site database not a cookie.