Would a SSD be faster for many small reads/writes?
Quite. This is the whole purpose of solid-state drives - no moving parts, so virtually no seek time (see the question What are the pros and cons of a solid-state drive? for more details).
Are file system errors and bad sectors more or less of a problem with SSDs?
It all depends on your usage patterns, but I would say more. You still need to perform frequent filesystem checks, make sure your OS sends the proper TRIM commands, and use the drive manufacturer's software to continually monitor the health of the drive (SSD sectors have only about 10,000 writes before they become unusable).
Is there any reason not to buy a external SSD over HDD?
Yes, but that depends on your situation. If your hard drive is external, but not moving around much, I would go with the hard drive - it will probably last longer. If you need to move the drive around a lot (especially when in use), or have specific performance requirements (and can withstand the limited write cycles of the drive's sectors), then you may want to consider the solid-state drive.
Also, you may be limited by the connection method since it's an external drive. If you don't have eSATA, and your computer doesn't support USB3.0, then it's unlikely that you can use the SSD to it's full potential transfer speeds - and because of this, you might not notice much of a difference.