They're not programs. And they can't be played by any player (I think. That'd be weird...). I have one showing as a VLC player file and another as a Harmony assistant file, but that makes absolutely no sense and besides, lately my icons seems to have a mind of their own so I don't really pay any attention to that.
Anyway, as I suggested, these are not files to be tampered with unless you've backed them up and/or have no Incomplete dlds to resume. The only thing I would tell you to try is to just put the files you want (those incompletes from Kazaa) into the Incomplete folder and open LW to see if in the Library's incomplete folder they have (somehow) appeared.

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iMac G4 OSX 10.3.9
RAM 256MB
LW 4.10.5 Basic
ADSL anything from 3 to 8Mbps/around 1024kbps

"Raise your can of Beer on high
And seal your fate forever
Our best years have passed us by
The Golden Age Of Leather"
-Blue Öyster Cult-