You are correct, your OS will only boot up faster with a solid-state drive. That being said, if you perform any operating system related task that needs to retrieve data from the drive, it will be much faster then if your OS was on the HDD. That being said, if you only care about loading subsequent programs (or logistically prefer keeping your OS separate), you can easily keep the SSD as a secondary drive, only using it for certain programs/tasks.
The whole point of a solid-state drive is to decrease application loading times, and that's it for most users. This is more due to the lower seek time rather then the faster transfer rate, which makes it more like RAM (good sustained and random transfer rates). In fact, some users would be better off getting more RAM then a solid-state drive - but that always depends on your needs as the user.
That being said, the main storage device (SSD or HDD) is always the bottleneck of any computer system. While SSDs help to alleviate this bottleneck, new ones are still only ~1/40th the speed of RAM. For example, some memory bandwidth in newer computers has reached over 20,000 MB/s versus some new SSDs which top out at just over 500 MB/s.
You can also use it for the increased sustained transfer speed, but that only applies if you deal with very large file transfers with, for example, video encoding.
For the fastest experience with your computer, install your OS on the solid-state drive, but do remember to make frequent backups. Yes, it will mostly affect just your load times, but again, that's why you put data on a solid-state drive in the first place. They are not meant for storage of large amounts of data, just your OS/programs/games only.
Finally, since most SSDs have a smaller capacity then hard drives, you may wish to "lighten" your OS install by not installing as many packages (if you use a package-based Linux distro), or using a utility to remove components from the installation media (if you use Windows).
You ask why SSDs don’t necessarily improve the speed of running applications compared to a traditional HDD. The reason is disk access. SSDs do read and write much faster than traditional hard disks. However, unless the application is heavy on disk I/O (reading and writing to the disk), the benefits of the SSD are lost. Even in cases where there is some writing to disk, the OS has a disk cache. This means writes are written to RAM and eventually flushed to the disk at a later point. The disk cache significantly improves the speed when using traditional HDDs.
As for swapping, yes, SSDs will perform faster, but you probably won’t notice the difference. Unless you are severely starved for free memory and the OS is heavily swapping, you will not notice the speed difference.
Best Answer
No. While slow I/O can contribute to these errors, there are more reasons why they could appear.
Still, it's worth getting an SSD because the system will feel much, much more responsive.
That being said, 8 GB of RAM is bare minimum these days even for light Internet browsing, not to mention development and running heavy IDEs. When the RAM gets scarce the system will start moving less frequently used data to the HDD/SSD. At this point the speed of RAM doesn't matter because even fastest SSDs are orders of magnitude slower than RAM and that becomes the bottleneck.