Two things seem to be happening here.
First it looks like there are permission issues with the mysite folder in your installation. Check that your user (or the webserver user) have the correct permissions to the installation. Either your user or the webserver user will need to have read/write access to at the very least the following;
As for mod_rewrite and AllowOverride being set incorrectly, I have used that 000webhost before and have been able to setup Silverstripe in the environment with no real issues. I am using the free accounts, and will have a peek at my installation notes a bit later.
And if your install still does not work - although the installer says everything should be peachy: go into your silverstripe database and delete all the tables (note: all data in those tables will be erased). Then go directly to your install.php and run it again. Important: do not go into your main silverstripe dir or to any other part of that website or else silverstripe will recreate those tables again.
After that everything worked fine for me - and for other people as well.