Android box TX 3 pro - help needed!!

Perusing through that link you posted, the download for the firmware is archived as a .rar file. After downloading it, when you opened that rar file up, was the indicated .img file (LC-ALE-1-TX3pro-20170228.img) there?

Good to hear that the firmware flash process worked out after you tried again.
As for your queries, I don't have a TX3 Pro so just making assumptions in the hope that someone here in the Forum with actual hands-on experience will chime in with more definitive answers.

-- Generally when an app, especially one that manages a video streaming service, gets minimized into the background, the active video stream continues on as is.
-- This doesn't appear to be a custom ROM but rather that site was providing a copy of a stock ROM. Usually with a custom ROM that requires the device to be rooted, flashing a stock ROM however does not. As for its validity, that's hard to tell without actually directly comparing the two ROMs side-by-side. That site, China Gadgets Reviews, 'appears' to be legitimate but for anyone to determine whether any web site is or isn't legitimate anymore is a big, open question. I'd be leaning towards the firmware they provided is OK -- there's a bigger chance downloading a file from an individual hosting his/her own server is riskier than from a commercial venture. But again, there's no 100% assurance from one or the other, and determining just where GGR originally obtained that firmware is a mystery.
-- I would never rely on just one anti-virus/anti-malware utility if something is found during a scan. The chances of false-positives is actually really, really high and despite all the marketing B.S there is no utility in existence that can catch every exploit out in the wild. They typically have a success rate in the 40%-90% range, and that's always going to be based on a narrow time span (new exploits are popping up frequently and on a consistent basis so a utility may do great at detecting problems in one test but the next quarter, not so much).
-- As for that update message always popping up, try doing a Factory Reset (.. the caveat that backup anything you have saved as a Factory Reset will delete all user stored data). On a typical Android device, the internal storage is divided into several partitions -- most are dedicated to the operating system and one for general, data storage. I'm assuming (emphasis on that) your TX3 Pro is similar so the issue is with a Factory Reset, that only wipes that data partition clean. The big variable being different manufacturers set up their own products in different ways so for some, when flashing the ROM that wipes not just those system partitions but also that user data partition. But with some, like Samsung, flashing the ROM only affects the data partitions, leaving the user data partition as is. So when you re-flashed the firmware, that may or may not have also wiped the data partition. But if you initiate a Factory Reset on your own, at least that eliminates the possibility either way.
You might also want to try wiping the system cache partition. Again, not knowing just how your T3 Pro is configured, the flashing of the firmware may or may not have cleared the system cache partition, but it won't hurt anything to do it and can only help. But you do need to boot your device into it's Recovery Mode to do the 'wipe cache partition' option. Not familiar with a TX3 Pro so I have no suggestion on what the button sequence is to boot into Recovery Mode.

Not having any luck finding any definitive documentation on your TX3 Pro. Other than being well rated and getting good reviews it appears to a pretty good bargain ($30 range) for the feature set it includes. Did any kind of user manual come with it?
1. Are these the general hardware specs for your unit?

Android 6.0 OS
CPU:Amlogic S905X Quad Core Cortex A53 Processor up to 1.5GHz
GPUenta-Core ARM Mali-450
RAM:1GB DDR3
ROM:8GB eMMC
WIFI:IEEE802.11 b/g/n
LAN:10/100Mbps Ethernet
Support English, Russian, Hebrew, Portuguese, Spanish, etc., multi-language

VP9-10 Profile-2 up to 4Kx2K@ 60fps ,H.265 HEVC MP-10@ L5.1 up to 4Kx2K@ 60fps
3D, CEC, HDR and HDCP 2.2
HD MPEG1/2/4, H.265, 4K-HD, HD AVC/VC-1, RM/RMVB, Xvid/DivX3/4/5/6 , RealVideo8/9/10
Avi/ Rm/ Rmvb/ Ts/ Vob/ Mkv/ Mov/ ISO/ wmv/ asf/ flv/ dat/ mpg/ mpeg
4K@ 60Hz video decoding 10-bit HEVC and AVS+ codec support; H.264 up to 4K 30Hz
Hardward 3D graphics acceleration
MP3/ WMA/ AAC/ WAV/ OGG/ AC3/ DDP/ TrueHD/ DTS/ HD/ FLAC/ APE
HD JPEG/ BMP/ GIF/ PNG/ TIFF

Ports:
1*SPDIF
2*USB
1*HDMI
1*RJ45
1*TF card slot
1*AV
1*DC
With only 1GB of RAM that is a relatively low amount, but not bad for a single-purpose set-top box. The issue is Android's integral memory management is going to be more better than most of us are trying to do the same. Users micro-managing memory was a thing with Win98 and pre-OS X systems, not so much now. Most of the time we just end up creating a situation where the device is working that much harder to compensate for our actions. So killing background apps is good for our own peace of mind but it might not actually be optimizing RAM usage and making your device run better. Just be judicious about it.
2. Have you tried using an Ethernet cable to connect to your home LAN instead of WiFi? If that message is only popping up when using WiFi connectivity, try deleting the WiFi entry you added in the Settings menu and then add it back in again. You might also want to reboot your router, sometimes there's just some one-off networking glitch that will get cleared with a restart.
3. No comment, since there's not a user-interactive screen involved I don't know much granular control is available to you. Again, hopefully someone with actual experience with chime in.
4. If you're referring to some kind of anti-virus/anti-malware app that came pre-installed with your unit, the odds are you can't do much to actually uninstall it. That would require your device to be rooted. If there's any way to manage the installed apps in the Settings menu you might be able to disable it though.