Can Backup of ROM contain adware/virus

[Blue Sky]

Hello,
I have a Samsung Galaxy S4, Chinese cloned or fake, whatever is called, with the following info:
Gt I9500
Android version: 4.2.2
BUILD NUMBER: JDQ39.I9500ZHUAMDE
CUSTOM BUILD VERSION: 1378866812
baseband VERSION: MAUI.11AMD.w12.22.sp.v7.p3,
2013/04/11 12.04

BB chip: Universal 5410
MS board: Galaxy S4
Samsung Exynos Octa 5410 1.89 Ghz
cpu Renderer: Power VR SGX 531
Display: JDQ39.I9500ZHUAMDE
Hardware: mt6577
screen resolution id dp: 320 * 533
screen Density: 240 dpi

2 days ago it was infected with some virus/adware, called, netalpha , so far I could not get rid of it, I did factory reset few times, and using some tools to get rid of this nasty adware, but it comes back again,

now I want to flash the mobile to a good working custom ROM, or even the clean ROM that is already built, but I don't have the back up of it,
So I want to make a backup of the ROM of my mobile first,

my question is:
1- does the back up I take will contain the virus/ adware as well?
if so is there a way to take a clean back of the ROM?

2- my intention is to restore my mobile with the original ROM that is inside , is it possible? with the situation I have, because I am not sure if I can find a ROM that works with this mobile, since this mobile is not original.

3- To look for Custom ROM, what should I look for that is important?
CPU?, Build number? screen resolution?, baseband VERSION? or all these together and some others?
I have seen some Custom ROM for cloned Samsung Galaxy s4 I9500, which some of info matches like mtk6577 but not all of them, I am not sure what to look for.

appreciate your help