Hardware solution for live video conversion To briefly summarize: I am looking for a video conversion box that will accept any resolution/refresh rate/color space and split out a properly converted feed.

I utilize a hardware H.264 encoder/video switcher called the ATEM Television Studio for live-streaming company events. It's worked wonderfully for all intents and purposes, except it requires precise audio and video inputs. 

For audio, I use a Behringer DEQ2496 to convert my audio into a usable AES/EBU digital signal and to provide an audio delay (when needed). Super useful.

For video, I have a laptop that sits at 720p and I've set a 59 refresh rate--and it is accepted as a 720p59.94 signal. As such, it requires any other video input be precisely 720p59.94.

In the past, I've requested A/V companies I've worked with to supply the video signal at precisely 720p59.94. This has caused some issues, the last conference we hosted I was barely able to get working (they had to find a spare video conversion box on-site, as the first box they were using did not give a correct signal). 

I would like to take the A/V company out of the video conversion process and be able to accept any video input, and convert it. I would like to be able to accept both HDMI and SDI inputs of any resolution/refresh rate, and be able to convert that signal into something usable. 

I am looking at a Blackmagic Broadcast Convertor, but the lack of documentation about it has me slightly concerned. I'm not very familiar with any non-Blackmagic solutions for this problem.

Does anyone have any familiarity with the above equipment, or have a similar piece of a equipment they've used for live, on-site video conversion?
That or the UpDownCross converter should meet your needs as long as the frame rates are the same.  I'd personally stick with BlackMagic's hardware as you already have their ATEM and they make some of the most cost-effective hardware around.  They aren't the top quality brand for sure, but you won't beat the value for the price pretty much anywhere else.

If you need to change the frame rate, you will need some kind of frame buffer that can interpolate on the fly, which is likely going to get pricy.

Alternately, is there a reason that you can't vary the output of the laptop to fit whatever quality is needed for the feed from the A/V company?  This would avoid the need for a converter and seems to be the easier option.  A better output for the laptop may well be a cheaper solution.  The Intensity Shuttle for example is only $199 if you have a thunderbolt or USB 3 port.