So you recon that your connectivity to external hosts is fine but you struggle with the 'appbrowser': Have you tried a different mirror?
As I was preparing for this post I had a bit a "play" and to simulate the apparent inability of your system to connect to 'ibiblio' I used
sudo sed -i '$a10.0.2.2 distro.ibiblio.org' /etc/hosts (and
sudo sed -i '/10.0.2.2 distro.ibiblio.org/d' /etc/hosts to back out this tweak again). What I found is that you might be in a classical "catch-22" situation insofar that 'appbrowser' can't get the list of mirrors from 'ibiblio' and hence you can't select an alternative host in 'appbrowser'.
Before I give a possible solution I wonder what happens when you execute
tce-fetch.sh info.lst.gz (which is AFAIK pretty much the first thing 'appbrowser' does when you press the 'Connect' button). If that fails with something like
wget: server returned error: HTTP/1.1 404 Not Found I'd suggest you execute
echo 'http://ftp.nluug.nl/os/Linux/distr/tinycorelinux/' > /opt/tcemirror and repeat the test with 'tce-fetch.sh' again. If that works your 'appbrowser' should work as well.