I understand that you have some questions about the SlingCatcher and the Slingbox SOLO and their features.
1) Would faster upload speeds at home improve the picture quality?
Yes, the higher the connection speed the best the video quality. The Slingbox SOLO needs at least a connection speed above 600kbps in order to stream a good video signal. On the other hand, if you get a Slingbox PRO-HD, you will need to increase your current connection speed, since you will stream a HD signal. The following links will provide you with all the information about the requirements for both the Slingbox SOLO and the Slingbox PRO-HD:
2) Would an HD-capable Slingbox help? My video source is an HD cablebox/DVR. I believe the Sling catcher is not capable of show HD or hi-res images but I'm wondering because I want to purchase another box so that work could watch what they want without affecting what's viewed at home.
The Slingbox PRO-HD allows you to stream the HD signal as long as you meet certain requirements. As stated before, if you have in mind to stream the HD signal using a HD DVR, you'll need to increase the connection speed. Besides, the Slingbox PRO-HD allows you to split the video signal, without affecting to home viewers.
You need to set a splitter on the cable that comes from the wall. Then, you will send two lines: one will connect to your DVR and the second one to your Slingbox. The only disadvantage is that the Slingbox will get the basic cable service channels only.
Finally, I encourage you to check this link in order to get further details about the resolution you can get with the SlingCatcher:
Hope this helps.