[ONE Q&A, Help & Troubleshooting] Ctuiku:
I feel like i need to reanswer a some of your question even though some already have answered them. You seem very interested and it seems fair to clear up some things.

There is a little misconception about what cyanogenmod is.

The offical Oneplus One comes preinstalled with CyanogenOS (or CM12S), which is derived from Cyanogenmod. It get's a bit complicated. Cyanogen is a company founded by the lead programmer of Cyanogenmod. CyanogenOS is a commercial product, that is licensed to phone manufactures.

Cyanogenmod is the codebase for CyanogenOS. CyanogenOS adds some functionalities that are not present in the community maintained Cyanogenmod. (e.g. a alternate dialing app, theme store, etc.)

The offical Oneplus One comes NON-rooted. You need to manually enable root.

Please keep in mind that it gets REALLY complicated from this point!

The Oneplus One debuted with CM11S (CyanogenOS, the commercial version of Cyanogenmod). Cyanogen (the company) promised users 2 years of support and updates for the owners on the Oneplus One. But Oneplus and Cyanogen had some major issues that resulted in ending the partnership.

Oneplus had to offer an alternative Android Firmware for their upcoming phone Oneplus Two. So they did another android version called OxygenOS. This was installed onto all Oneplus One phones from a certain date onward!

THIS is the OFFICAL Android Software Version for your Oneplus One!!!

See: https://oneplus.net/de/support/answer/how-can-i-download-oxygen-os

I would strongly recommend to install OxygenOS, as it is the easiest way of experience the phone as Oneplus intended it to be.

Most tutorials on here are a little bit older so they are offering guides which mention either CM11S, CM12S, Cyanogenmod or OxygenOS. (or if you are more familiar with other customroms, whatever you'd like to install! But this is something you should look into, when you are more familiar with what costumroms are and what they do! There are so many excellent guides on here, which i'd recommend you to read sometime!)

He rooted it either manually or used an android version (e.g. something downloaded off XDA or got one from some developer). Such versions are often called Customroms.

Also note, that there are no official retailers of Oneplus phones. They are officially offered from the website only. However, there are some shops in China, that offer Oneplus phones.

Most phone manufactures actually tell you that your warranty is void when rooting your phone.

Oneplus however is very, very supportive when it comes to rooting!

And of course installing other Android Versions and Roms!

See: https://oneplus.net/de/support/answer/will-rooting-or-unlocking-the-bootloader-void-my-warranty

Cyanogenmod is indeed a custom rom. (or sometimes aftermarket firmware)

See: https://en.wikipedia.org/wiki/CyanogenMod

A stock rom is the firmware that is installed by the manufacturer. (in your case: CyanogenOS, called CM12S. the 12 stands for the version number of Cyanogenmod. 12 = Android 5, 13 will be Android 6. There was a CM11S version, when the phone debuted last year.). The S marks it as CyanogenOS, the commercial product. So CM12 would be the community maintained Cyanogenmod. CM12S the product by the company Cyanogen. As said before, it's a bit complicated, as the current OFFICIAL rom is OxygenOS

Your findings of software not present in the official stock Oneplus One rom (which is either CM12S or OxygenOS), suggests your software on the phone is actually a unofficial costum rom.

See the answer to the rooting question. Almost all major phone manufactures void your warranty if you flash a custom rom. As it involves tampering with system files and most people are not very knowledgeable when it comes to "computery things". It's just a security measure, so that people won't sue the company when something bad happens (most likely deleting precious pictures, etc.)

Oneplus actually encourages people to try other roms. This made the Oneplus One a very popular phone for tinkerers and as you can see on this forum there are a lot of custom roms you can choose from.

If the seller was not so careful, there will be a notice, which can be read by software programs available here on XDA, that say how many times a custom rom was installed.

However, i think your main concern is, that you wouldn't be able to get support form Oneplus, because you think your warranty is not valid anymore. As said above, your concerns are not that problematic. Especially as you seem to be knowledgeable enough to actually fix things on your own! :)

A factory reset will delete all your apps and reset the system to the state"it came out of the factory". So when you start up your phone, you need to set it up again. Enter your name, configure your WiFi settings, install your favorite apps.

However. Your photos and your music files will not be deleted! Be careful though. Resetting your system will delete your app settings. I just want to point that out, because the first time a did a factory reset and reinstalled everything i thought i lost all my Whatsapp photos. They were there in the Picture folder, but weren't linked to the images in my chat window! This is a bit complicated, but can be solved quite easily, if you know what your doing (backing up your apps with certain programs, e.g. Titanium Backup).

Factory reset is a internal function of your phone (to be precise of your phones recovery!)

However, as most people said and your suspected is: Your phone does not run the official factory image, provided by Oneplus (or Cynaogen).

The official software can be obtained from

https://oneplus.net/de/support/answer/how-can-i-download-oxygen-os

Yes it will most likely to what you fear. It depends on what the retailer did. Most costum roms include software in their phones, that will be installed again with a factory reset. If your seller did install the programs AFTER the phones original state (e.g installing it from an appstore or from the APK's (Androids Software Packages)) they won't be present!

Absolutely! I'd recommend you to flash a new image to your phone!

There are a lot of great programs here on XDA that offer easy and intuitive options of doing that, without even knowing anything about flashing programs! Try to take a look around and see what you can make of it.

I personally think they are great, but can be dangerous if you didn't read the proper instructions. I'm just saying, as they often delete EVERYTHING on your phone. So whatever you do to your phone, backup everything before!!!

But these simple programs are what you are looking for. They will install the firmware (Cyanogenmod or OxygenOS) that are bloatfree and as the Oneplus intended them to be.

This further suggests, that your are not running an official version. The official update process is fully automated and doesn't need any user intervention! From your description it doesn't do anything and you are stuck in recovery, that is waiting for user input. This is not the case with proper official roms!

Absolutely correct! See above.

You do not need root to install a recovery. This is a bit complicated, as there are some misconceptions about what is what. You need to "unlock" your phone to install another recovery. This is not rooting, but is also restricted by many manufacturers. But as with rooting, Oneplus is offering support on unlocking your phone and there are lots of guides on how to do it.

Wise decision.

I would strongly recommend to NOT leave it this way! As you yourself found out, there are chances that there are programs on the phone, that are not to be supposed to be there. I'm not saying that there is spyware on your phone, but it is most likely that there could be malicious code running in the background of your phone. I would strongly recommend NOT to use apps, that need sensitive information of yourself!

Try to look for guides and tutorials here on XDA and just install CM12S, Cyanogenmod or even better OxygenOS, the official ROM from Oneplus! OxygenOS is not everyone's favorite choice, but that way you can be sure, that NOTHING fishy is going on on your phone!

There is an easy app available from the offical oneplus site!

https://oneplus.net/de/support/answer/how-can-i-download-oxygen-os

Or use the app that is mentioned in the post above! Will to the same thing!

Use the instructions on there and you are much safer on using your phone than you are in the present state!

BUT PLEASE, PLEASE DO A BACKUP OF YOUR DATA! I mean, your photos, contacts and everything important before doing anything to your phone. I'm just stressing that out, as so many people complain afterwards that they lost everything, etc.! Not just because the process they did was unsafe, but becaus the didn't properly read the instructions that tell people that they should make a backup, BEFORE tampering the system. It says factory RESET, so it should be clear, that everything will be reset!