I don't see any reason why the D521 wouldn't be 7 compatible. I've upgraded Vista PC's to 7 before without issue (aside from rare driver issues that are easily fixed). It depends on the version of Vista you are upgrading. For example, you can't do a straight upgrade from Vista Basic or Home Premium to Windows 7 Pro without a clean install. You can do it from Vista Business just fine. Vista Home Premium goes easily to 7 Basic and Home Premium. The hardware of the D521 should be compatible with 7's legacy drivers. You'd have to install some drivers for the GPU, audio, display adapters, and storage or any media drives, but otherwise you should be fine. The fact that it's asking you to choose from two OS's at boot is a flag. Obviously there's a damaged or missing boot file. The fact you aren't getting any kind of boorldr or bootmgr error is odd, but it's probably due to an incorrectly executed upgrade or an issue during the upgrade process. I'd just do a clean install.
You can retrieve the files you want without an OS, but you'd need to mount the hard drive with a SATA to USB adapter, or you could do it by using another desktop PC. Shut down the computer, install the hard drive into a free SATA port on the motherboard and a SATA power connector. After rebooting the PC, click the Computer icon and check to see if your old drive is visible. Access your drive's files through the host PC's OS and copy over the ones you want to save. You could even image or clone the drive.