My Thinkpad Edge E440 is one of the most basic models (20C5A012SG) without NVidia graphics(I only have the GPU on the Intel i5 4th generation processor). The WIFI interface installed on my machine is an Intel Wireless-N 7260. The following devices work from a fresh Slackware64 14.1 installation:
USB 2.0 and 3.0 ports
Touchpad
Web cam
Display in X.org
Gigabit Ethernet port

The only notable piece of hardware not working is the WIFI interface. To make it work, I compiled a more recent 3.10.28 kernel:

1] I've disabled UEFI in the BIOS. I don't use it.
2] Install Slackware64 14.1. The only disk sets I do not install are E, F, KDEI, T, and Y.
3] Update Slackware64 14.1 first. Use the Gigabit Ethernet port for internet access.
su - # Or login as root.
cd /etc/slackpkg
cp -p mirrors mirrors.orig
vi mirrors # Enable a mirror.
slackpkg update
slackpkg upgrade-all
shutdown -r now

4] Do the following to compile the Linux 3.10.28 kernel:
su - # Or login as root.
wget https://www.kernel.org/pub/linux/kernel/v3.x/linux-3.10.28.tar.xz
cd /usr/src
tar xJpf ~/linux-3.10.28.tar.xz
rm linux
ln -s /usr/src/linux-3.10.28 linux
cd linux
make mrproper
cp -p /boot/config-huge-3.10.17 .config
make oldconfig
make menuconfig # You may skip this. Do this only if you need to make configuration changes.
make bzImage
make modules
make modules_install
cp arch/x86/boot/bzImage /boot/vmlinuz-huge-3.10.28
cp System.map /boot/System.map-huge-3.10.28
cp .config /boot/config-huge-3.10.28
cd /boot
rm System.map
ln -s System.map-huge-3.10.28 System.map
rm config
ln -s config-huge-3.10.28 config
rm vmlinuz
ln -s vmlinuz-huge-3.10.28 vmlinuz
/sbin/lilo

5] By default, sound won't work correctly. To correct this, create an /etc/asound.conf with the following content:
pcm.!default {
type hw
card 1
}

ctl.!default {
type hw
card 1
}

6] Restart your computer:
shutdown -r now
7] Login as root. And then adjust your ALSA mixer settings:
alsamixer
8] Save your ALSA mixer settings:
alsactl store
9] Congratulations. Your Thinkpad Edge E440 with Slackware64 14.1 should work fine now.