These are text-formatted, line-deliminated files written by applications like mysql or apache that are out of my control.
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58answers:
3If they're all networked and on the same domain; make a shared folder, and have the programs output their log files to this folder.
It really depends on what you are comfortable working with.
You could perform this task with anything ranging from scripts (bash, perl, .bat, etc.) to compiled programs you write yourself to various products (Splunk and Q Radar were mentioned by tschaible) or even something like Microsoft's SSIS.
Hard to say much more until we know what platforms/languages/etc you like to use.
Most likely you would use FTP as a lowest common denominator for pulling the files from the various servers. It can be used by all the options I mentioned.
Seems likely you would want to pull them into a database of some sort to make them searchable.
You could probably get a better answer if you add some more details to your question.
Hope this helps.