I want to use a Raspberry Pi (RP) as an SSH server, but I don't have an HDMI or RCA monitor/TV to use with it. It is possible to convert the HDMI/RCA output of the Raspberry Pi to USB that can be read by my computer (a laptop) so I won't need a monitor or TV?
Once the initial setup is complete, I'll be able to remote into the RP and finish my setup remotely.
Converting either the (HDMI) digital video or (yellow "RCA") composite video to anything else (e.g. VGA) is going to involve a converter box that is going to cost more than a serial port connection. There is simply no converter for video out to USB into anything besides video capture devices.
You're asking an XY question.
What you really want to accomplish is to use something besides a video port as the console.
The simple solution is to use the RP's UART as the console, and then use a TTL converter to RS-232 or USB to connect to PC/laptop. Directions are here and here and here. An all-in-a-cable assembly is this USB converter (photo is below text). A UART for the console is a common/standard procedure for bringing up embedded or single board computers.
The serial port connection has the added benefit of consolidating input and output so that you no longer need to attach a keyboard and mouse. The downside is that you now have a text interface rather than a GUI; however that is typical for a server.
Another solution is to use a networked console over Ethernet.
