I believe my iPhone is jailbroken

Since the release of iOS 10, the ability to delete some built-in apps has been available. Remove built-in Apple apps from the Home screen on your iOS 10 device or Apple Watch - Apple Support

If you are not seeing AT&T in the upper left corner, what are you seeing? Has anyone had possession of your phone that could have jailbroken it?

EDIT:

The image that you just posted is one that is received if you do not have the latest version of iTunes installed on the computer you have connected the phone to. If you go into iTunes>Help>About, what version number do you see?

the current version of iTunes is 12.7 you appear to be a few versions back which may be indicative of an earlier version of OS X, possibly 10.7.x, in which case you may need to update you OS X to facilitate the install of iOS 11 through a newer version iTunes.

Apple suggests you update to the current version of iTunes if you plan to use it to update your device.

Update the iOS on your iPhone, iPad, or iPod touch - Apple Support

The legacy of your OS may be attributing to any issue with the OS X legacy version of iTunes you have installed working properly with iOS 11.

Apple no longer support OS X prior to 10.11, anything working was likely not tested for compatibility and just happens to still work; anything not working may be due to incompatibility with the features inherent to that OS.

if you need to do this you should backup your system and install the latest supported OS for your computer, provided it meets the requirements for that OS.

Use Time Machine to back up or restore your Mac - Apple Support

MacOS Sierra - Technical Specifications

if your computer does not meet the specifications of Sierra you can still download El Capitan at this time

OS X El Capitan - Technical Specifications

You have given us very little information that points to your phone being jailbroken. However, if you are still concerned, restore your phone to factory state:

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

The_Old_Man wrote:

And another thing is some of my default apps are deleteable. For example: Mail, Notes, Calendar,...

Look back at the first link that I provided you, which explained about removing built-in apps in iOS 10. That explains the ability to remove selected apps, including those you listed.

This also indicates the Mac requirements for the latest version of iTunes. It requires OS X 10.10 or higher, so your OS X is out of date to run the latest version of iTunes.

Mac System Requirements

Hardware:

• Mac computer with an Intel processor
• To play 720p HD video, an iTunes LP, or iTunes Extras, a 2.0GHz Intel Core 2 Duo or faster processor is required
• To play 1080p HD video, a 2.4GHz Intel Core 2 Duo or faster processor and 2GB of RAM is required
• Screen resolution of 1024x768 or greater; 1280x800 or greater is required to play an iTunes LP or iTunes Extras
• Internet connection to use Apple Music, the iTunes Store, and iTunes Extras
• Apple combo drive or SuperDrive to create audio, MP3, or back-up CDs; some non-Apple CD-RW recorders may also work. Songs from the Apple Music catalog cannot be burned to a CD

Software:

• OS X version 10.10.5 or later
• 400MB of available disk space
• Apple Music, iTunes Store, and iTunes Match availability may vary by country
• Apple Music trial requires sign up and is available for new members only. Membership automatically renews after trial

The_Old_Man wrote:

Forgot to say this, but I'm using iPhone 5. And what I'm seeing in the top left corner is 5 dots with the name of the Carrier.

The name of what carrier? What carrier SIM do you have in the phone? What makes you believe you should see AT&T. Please provide as much information as you can at one time, so we don't have to go back and forth with questions/answers. What troubleshooting have you done?

The latest version of iTunes is 12.7. What computer operating system are you using? And please, rather than editing your posts like you are doing, use the reply button to provide additional information. It is getting difficult following your posts.

The_Old_Man wrote:

VN Mobifone is the name of carrier. Also I read from TechWalla that if you don't see AT&T, you are jailbroken.

You will only see AT&T in the upper left if AT&T is your carrier. If VN Mobifone is your carrier, then everything is correct. Now, what other symptoms are you seeing that make you think something is wrong with your phone?

That may be the case if you have an AT&T iPhone with an AT&T SIM card.

Is AT&T your carrier? If AT&T is not your carrier you certainly should not see AT&T on your iPhone.