1) VPS hosting is a type of hosting where a specific amount of memory, disk space and/or processor power has been assigned to you. These resources are used to build a virtual small computer inside a larger server. Such a virtual computer can run a large range of operating systems, mostly Windows or Linux and the Operating System itself is not aware of the fact that it isn't running on a physical computer, but inside a process which is running together with a lot of other isolated virtual servers on the same physical computer.

The good thing of VPS systems is that the hosting company can easily change the resources you are allowed to use, i.e. amount of diskspace, amount of RAM to be used etc. With shared hosting all processes eat from the same central memory, disk and CPU resources and if one process peaks in usage of CPU or memory, there is less left for the other processes. With a VPS this can be much better controlled.

2) The huge spike could have several reasons, including a visit of a scraper to your site, a periodic backup process which started or it could have been made up by the host just to force you in a more expensive hosting package to increase earnings.

3) If done right, there is only a little downtime involved and the search engines will pick up the new location of the site within hours, or maximum a day or so.

You should understand that a VPS is not the solution for all problems. In fact a shared server environment as you use now where your processes can use a large amount of RAM and have a large amount of CPU power available is much better in gracefully processing peak usage than a VPS where the amount of RAM is often limited to some low-end value like 512MB. With low memory availability your web processes will simply crash if they reach the upper memory limit. The hosting company won't complain, but you certainly will. Especially sites which do not have a smooth usage pattern can suffer from this.