Is it dangerous to change or remove company-brand boot logo? There are two types of boot animation:

Company-brand logo
ROM-based

I was able to change ROM-based animation by rooting the phone and using a custom ROM.
However, I'm a bit confused about how to change or remove the brand logo animation. Is it safe to do that? Will it increase the speed of the booting process?
Yes it is possible to change the company logo. If you have ever flashed a custom ROM which changes it and still have the update.zip file lying around, sometimes (after extraction) you will find a logo.bin (its a raw Binary file) file in the root directory. This file contains the Brand Logo as well as the charging animation and the image to be displayed in fastboot mode. However on your phone (and most other phones) there is no logo.bin file but just a partition named LOGO where a complete dump of the logo.bin file is saved.

There are update.zips just for changing your phone branding, however only files for the correct resolution and chipset will display correctly. (Yes, you can indeed use such files made for other phones with the same SoC with the same resolution)