1. No you do not have to be online to install pre-downloaded updates
2. If you are getting failed updates, it could be due to a number of reasons.
In your case, it sounds like the hard drive is at fault. A faulty or slow hard drive will bottleneck installs like this and result in failed updates. The checksums won't match or files won't write properly due to hard drive problems.
With the computer being slow overall, I can't imagine it being a separate issue. If you have a 3rd party anti-virus like McAfee, Symantec, Norton, AVG, or whatever, you need to make sure its disabled while performing this process. Anti-virus products interact with the hardware abstraction layer on the computer, and if written poorly (most of the pre-bundled ones are), you will have difficulties.