[Brand new server errors]

Can't see any errors, it seems to be loading fine. The large amount of text at the end is normal, it's TreeCapitator that does some "stuff"

Have you tryd to enter the server?

[Tekkit lite error]

Hey

"Some files failed to download or were otherwise corrupted."

Download http://files.minecraftforge.net/fmllibs/argo-2.25.jar

Download http://files.minecraftforge.net/fmllibs/guava-12.0.1.jar

Download http://files.minecraftforge.net/fmllibs/asm-all-4.0.jar

Download http://files.minecraftforge.net/fmllibs/bcprov-jdk15on-147.jar

Add them inside the "lib" folder at "%appdata%\.technic\tekkitlite", replace any existing files and try agian.

[Server Lag Problem]

Yes, you set the memory limit in the launch.bat to 1.2 GB and that might be a bit low.

Since your rented server had 4GB i will say it safe to set the limit to 3GB unless you use the server for something else aswell.

java -Xmx3G -Xms2G -jar TekkitLite.jar nogui

Try that in the .bat file and see if it works better.

[Server Lag Problem]

Hey

I personally think the ram is to low and that might be the cause of your lag problem and why you can't move so far from base, since the server struggle to allocate the memory needed.

We are 2 people on my server and it eats up 2500m atm.

For the color problem, i read that the Modular Powersuits - Tinker Table was makeing the signs text white and some other texture got like a shadow over them. I updated the Modular Powersuits mod to 0.3.0-178 and the signs returned to normal.

[Optifine and Tekkit Lite]

org.lwjgl.opengl.OpenGLException: Invalid value (1281)

Did read somewhere that Error 1281 probably means "value out of range" or something like that, generic to textures/material parameters, etc.

Can you try without the texture pack?

[Texture pack for Tekkite Lite?]

BDcraft have a tekkit lite texture pack (16x 32x 64x 128x)

http://bdcraft.net/forum/128x64x32x-tekkit-lite-smp-050-addon-pack

http://bdcraft.net/patches-list-search?m=2 (List that shows Lite, Classic and Voltz)