The problems you are encountering may be due to cookies or JavaScript being disabled in your internet options. Please exit out the launcher and try the following steps:
1. Open Internet Options
Open Control Panel → Select Network and Internet → Choose Internet Options
2. Click on the Security tab and select the Custom level button
3. Scroll down to Active scripting. Select Enable, then click OK.
4. Click on the Privacy tab and then click the Default button if it is available.
5. Click OK and then open the FINAL FANTASY XIV launcher again.