Welcome DW!

I've never read of putting the compressor in the basement here on this forum, but it sounds great in terms of an easy way to monitor it and protect the filter from getting dirty! Cold air won't be an issue as the bubbles of air can't change the temperature of the water around them due to differences in specific heat (heat carrying capacity) of air vs water.

Someone might chime in on whether there is any appreciable moisture put into the house from the compressor. Also you will want to think about keeping it cool and they do make some noise which might make you want to wall it off somehow or insulate from noise (just don't insulate the heat into the enclosure)

You can buy an out of the box diffuser head setup, or in the forums there are several pictures of folks who made their own base, used standard PVC pipe and fittings and bought just the membrane/disk to create the bubbles.

You often can do a craigslist search and find a GAST compressor or a rebuilt aeration compressor in your region. Search for 'aerator'

I'm not familiar with blue diamond pumps.