1.02 is the current firmware version. Repeater mode worked fine for me with a PC - well, "fine" other than it dropping the connection to the hotspot every 10 minutes or so. I doubt that Apple vs. Android matters.
When you use repeater mode you'll first connect using the default SSID as indicated on the included card (and on the DIR-505 itself), unless you changed this. Once you are associated with that network, open the browser, go to http://dlinkrouter, and the login page should appear. You can then use the wizard to set up repeater mode, selecting the "real" network. When the DIR-505 reboots, you'll have to go back to your WiFi settings on your device and connect to the new SSID.
I generally prefer to uncheck the box "Use the same Wi-Fi Network Name for the Extended Network if I am at a hotel or other place where others I don't know might want to connect to the SSID. If it's just in my home, then I'll leave the box checked. With it unchecked you get to specify a different SSID and password.
Also, note that the DIR-505 will not accept connections for 2-3 minutes after the LED turns green. Be patient. If your device tells you that it can't join the network, wait a minute and try again.