No logging whatever so e.g. encrypted servers, no tracking of originating IP address, etc
Most VPN providers will do only the minimal logging. Ask the support staff for exact details before signing up
Minimal or next to no impact on speed
The closer the server is to your location, the faster the speed will be. VPN's have a speed impact due to no cache on encryted data. Some VPN services, such as Astrill.com include a system in their software that recreates the cache localy (if turned on) speeding up the system slightly.
The ability to use the service on multiple computers without having to purchase a new router
Most VPNs are simple software installs on your PC. Usually the end user agreement allows you to install on multiple computers as long as only one is used at a time. If you want to use multiple computers on the VPN at the same time, you may need to buy additional keys to access the network. Again ask the support staff as this may vary.
Unlimited downloads and uploads
There are many VPNs who offer unlimited traffic. Look on their websites or ask the support for details.
Works with all web related applications e.g. web browsing, video streaming, chatting, audio streaming, etc
All VPNs I've tried work with the services you listed.