I don't know how to handle alpha channels...

LOL

How about follow a guide or something, or just trial and error like most people?

Which program are you using to edit files?

In photoshop you just open the converted dds image, switch to "channels"
palette (should be in the same "layer" submenu palette, if it isnt just
go to the drop down menu window->show->channels).

You can activate \ deactivate the alpha channel aswell as the RGB channels.

Alpha channels work in a greyscale where black is not-see trough, white
is completely see trough.

In the TYRE.dds theres a alpha embedded (coded?) in the game so you dont
need to actually draw the alpha channel, just paint the tyre thread in black
greyscale, what is white will be invisible what is black will be black, everything
in between will be mixed with the original LFS wheel color (so if its 50%
grey, i'll mix that 'darkness' with the standard beige wheel in LFS).

For other alpha channels encoding (DXT1a \ DXT1x \ DXT3 \ DXT5) they
all have the ability to hold a alpha channel, what differs is the bit of the
alpha channels that are being given, and the compression.