I've been using LW for over a year on my desktop computer...which crashed a few weeks ago. I have now bought a laptop, but wonder if I have to pay to download LW again? Any way to transfer it? I would love to be able to keep my LW Library tunes I've gathered.

You were using LW Pro or Basic? Basic is free to download.
If pro, then if over 12 months your support with updates will have finished. Search to see if you kept the Pro installer. That's all you need.

Transferring: More than one way to do it.
1. (a) Make a backup of your incomplete folder & LW download folder (Saved).
(b) Make a backup of the Downloads.dat file found in the LW preferences folder. (see Birdy's post here about where to find the Downloads.dat file Back up your Downloads file ...)

2. Install LW on the new computer then close the program.

3. (a) Then replace the new incomplete folder & Saved folder with the older ones you backed up.
(b) Replace the new downloads.dat file in the new preferences folder with the older one from the other computer.

* Keep a backup of the Downloads.dat file just in case.

One problem may occur with the Downloads.dat file in that it will have a record of the actual location of the incomplete folder. If this differs to the new computer then this is an issue. The downloads.dat file can be manually edited if necessary. If you are using the same Windows OS version then this problem will hopefully not occur.