Inspiron 580, can I run Windows 7 32bit OS disc on 64bit?

You will lose performance doing so and restrict the amount of RAM your system can use to 4 GB. You are best to clean install Windows 10 Home using your Windows 7 Home Premium Product Key. Windows 10 will run okay on this system if it has a Solid State Drive installed:

https://dellwindowsreinstallationguide.com/windows-oem-faqs-and-downloads/

The above guides also instruct in Download the Dell Windows 7 Professional Skylake Reinstallation ISO and making a Dell Windows 7 Professional Skylake Reinstallation USB. Instructions are given to convert it into a Dell Windows 7 Home Premium Reinstallation USB which you can use with your system. Windows 7 however reaches End of Life in January 2020 meaning it will have no more security patches and be no longer safe to use.

Thanks for your response. But, this inspiron 580 does not have SSD installed.

Doesn't matter whether it's ssd or spinner. Yes, you can run the 32 bit OS on the machine but as Philip_Yip pointed out you won't be able to access more than 4GB RAM

Theoretically you should be able to run Windows 10 without the SSD. In practice, the reason I mentioned SSD is that Windows 10 tends to elevate disc usage to 100 % with mechanical hard drives and you'll get a lot better performance with a SSD.

Thanks. Even if I have 8GB of Ram installed. Guess i don't understand enough to make an intelligent decision. The reason I even mentioned the win 7 ultimate, is because I have the OS disc. I didn't want to spend money for win 10. If I were to obtain win 10 from an ISO download, would I still need a product key, or would my machines key be sufficient?

There are no issues with the 580 and windows 7 32 bit EXCEPT that 32 bit windows 7 only supports 4 gigs of ram Maximum.  If you have a legit non pirated key you could use an OEM System Builder DVD copy of windows 10 1809 and tell it that you have windows 10 pro on install and that you do not have a key.   If you have a machine with 10 and have the media you can use that media to install.  Otherwise you have to buy or make an installer DISK or USB flash drive.

After Installing and getting online you run SLUI 3 from admin command prompt to use the win7 key to activate.

Iv'e added an extra hard drive to my XPS 8500. Running win 8.1 64 bit. I would like to know the procedure for booting to the slave drive, so I can install win 10, which I downloaded the ISO file and burned the disc.

Incidentally, I took your advice regarding my Inspiron 580. All is well and good with that PC.

Thanks very much.

Thanks for your help. My Inspiron 580 is running well with win 7 premium.

I have a new second hard drive installed in my XPS 8500 win 8.1. I guess it is referred to as a slave drive. I want to install win 10 on it. Can you tell me what procedure to use when installing the win 10 OS disc I have. Do i have to boot up to the second drive? any help with this procedure would be greatly appreciated.

There is no more MBR nor is there a master / slave drive.

BIOS is not a boot manager for windows 7 or 8 or 10.

You install a drive on SATA PORT 0 and install the os.