My question is whether 32 bit SharePoint Designer 2010 will work with SharePoint Server 2010 or not. SharePoint Server 2010 is 64 bit only while SharePoint Designer is available in 32 bit or 64 bit. Which one is best to work with SharePoint Server 2010?
The answer is yes, it will work. I think your choice will depend on what version of MS Office is already installed. My understanding is that they should match. So if you have 32-bit MS office, install 32-bit SPD otherwise 64-bit.
The simple fact that you can download a 32-bit version of designer tells me that it should work with SharePoint Server 2010.
As regards which version would work "best", I would follow Ashish's advice around currently installed products.
From the Microsoft download site for Microsoft SharePoint Designer 2010 (32-bit):
"This is the 32-bit edition of SharePoint Designer 2010. It is the recommended configuration for most users."