I understand that you want to know if you can hook up more than one Slingbox to your home router.
It depends on your router / modem capabilities and available ethernet ports. If your router shows at least one available ethernet port, the answer is yes.
Since you are already using the port 5001 for your Slingbox PRO, you only need to connect the new Slingbox SOLO to your router and create the rule using a different port number (5002, 8080 or 443 are recommended). Then, you need to run the same process as you did before with your Slingbox PRO in order to allow the communication between the Slingbox SOLO and the Internet. You will only need to pay attention to the available bandwidth since it is very important in order to keep a good signal.
Regarding the second part of your post, it does not really matter if you get a new computer. You can log in to your Sling Account from any computer with Internet access and that meets the requirements for the Slingbox SOLO. Then, you'll need to gather the Slingbox ID in order to add it to your Sling Account. Once you add the second Slingbox to your Sling Account, you will be able to get the connection from any computer, as long as it meets the requirements for the Slingbox SOLO. I encourage you to check the following links in order to get further details and information: