I am a college student. I only made $1050 last year for an internship but that was on 1099-NEC. Am I supposed to file (as my income is less than min) and pay tax/fee?

The 1099-NEC was created to report nonemployee compensation.  Nonemployee compensation is generally reported on Schedule C subject to employment tax.

If the income you earned was a "fellowship" and you were working for the pay, then it may need to be filed on Schedule C of your return.  

The filing requirements for self-employment income require a Sch C to be filed if self-employment income of $400 or more are received during the year.

Beginning with tax year 2020, the IRS will require your self-employment income of $600 or more to be reported on Form 1099-NEC (non-employee compensation) instead of Form 1099-MISC.

If the terms are that it is payment for teaching, research, or other services required as a condition for receiving the scholarship, then it is probably taxable as pay for services provided.

Use Worksheet 1-1 from IRS Publication 970 to figure the amount of a scholarship or fellowship grant you can exclude from gross income.

Once you determine the taxable portion of your fellowship stipend, then the following article explains how to enter it in your return.

How to report a fellowship stipend 

The IRS explains that if the following four conditions below are met, then the payments must be reported as non-employee compensation: 

1. The payment was made to someone who is not an employee
2. The payment was for services in the course of trade or business (including government agencies and nonprofit organizations)
3. The payment was made to an individual, partnership, estate, or maybe a corporation
4. The payments to a person totaled $600 or more during the year 

 Also, a business may need to file a Form 1099-NEC if they withheld federal income tax under backup withholding rules even if the amount if less than $600.

Yes you have to file.  The $12,400 filing threshold does not apply to people with a 1099-NEC. With an 1099-NEC. the filing threshold is $432 (the $600 is the limit for issuing a 1099-NEC).

The $148 you're paying is not income tax, it's social security and Medicare tax in the form of "self employment tax". 

As to the fee, 

TurboTax also has another free product called the Free File Edition (not to be confused with the regular Free Edition).  It has free Federal preparation and free State preparation. There is no efiling fee. It is not available after October 15

The Free file Edition can handle a 1099-MISC.

The Free file Edition is more fully-featured than the regular Free Edition.  The Free File Edition can prepare Schedule A, C, D, E, and F that the Free Edition cannot. 

To qualify for free 2020 federal and state tax returns through the IRS Free File Program delivered by TurboTax, you just need to meet one of these requirements:

• Your 2020 household AGI is $39,000 or less;
• You qualify for the Earned Income Tax Credit (EITC); or
• In 2020, you served as active duty military (including Reservists and National Guard) with a maximum AGI of $72,000, and you have a military-issued W-2.

The FREE FILE Edition is located at:
https://turbotax.intuit.com/taxfreedom

How to switch and start over in Free file Edition
https://ttlc.intuit.com/questions/2026912-how-do-i-switch-to-turbotax-freedom-edition