Oh, I just spotted a mistake. I forget that the PHP files are in the root of the folder, and shouldn't include the path being accessed.
Replace the relevent htaccess stuff with this corrected version.
~~~
# a folder
RewriteCond %{REQUEST_URI} ^/folder/[^./&]+/$
RewriteCond %{DOCUMENT_ROOT}%0 -d
RewriteRule ^folder/(?:([^./&]+)/)?$ /folder/index.php?path=$1 [QSA,L]
# a thread
RewriteCond %{REQUEST_URI} ^/folder/([^./&]+/)?([^./]+)$
RewriteCond %{DOCUMENT_ROOT}%0.xml -f
RewriteRule ^folder/(?:([^./&]+)/)?([^./]+)$ /folder/thread.php?path=$1&file=$2 [QSA,L]
# an index RSS feed
RewriteCond %{REQUEST_URI} ^/folder/([^./&]+/)?index.rss$
RewriteRule ^folder/(?:([^./&]+)/)?index.rss$ /folder/rss.php?path=$1 [QSA,L]
~~~