Without proper diagnostics and tear down it is impossible to say 100% what is at fault, however one strong candidate is the battery has malfunctioned, this can also happen when they go totally flat without recharging for long periods. When batteries malfunction they tend to disrupt the power up sequence. Unfortunately your battery is non removable. If you are handy you could open it up and unplug the batteries from the board and try it again.

USB RAM cannot compensate for physical RAM because the USB data transfer speed is very slow, especially considering the technology of USB your laptop uses it could even be slower than the hard disk. Generally speaking the laptop is a relic for archaeologists to find. That being said there are some things you can try to boost the performance a little bit, but do not expect miracles and even running a browser today consumes a lot of resources as such old machines struggle a lot. So what I recommend doing are the following. Preferably in the order I write the steps so you won’t need to go back and forth all the time 1: First off I strongly recommend starting with a clean slate and format your PC. If that can’t be done then run an antivirus antimalware program to make sure your system is clean. 2: Disable AERO themes in windows 7/8 3: Uninstall unnecessary programs and free up disk space 4: Disable programs from starting up with your PC that are not needed. Most programs have such options but they do not type...

I have a customer's Vaio that does that thing exactly, In this case it is definitely not heat issue, battery, power, or hard disk. I am in the process of running some other tests off windows, such as CPU, GPU and RAM stress tests but i seriously doubt that would be the case. My suspicion is drivers, even though i have disabled the one's that haven't been installed, from other forums I saw it could be GPU drivers, so try uninstalling your GPU driver and re-installing it. As a pro I cannot go directly to that solution without checking that everything is in perfect condition, but try it and let me know :)