Get your taxes done using TurboTax

Good!  I will attempt to enter data.  I hope that what I am doing is correct, but I cannot give you an absolute guarantee.  I believe it gets a satisfactory result.

     When entering data into TurboTax for W-2 form's Box 12 letter codes, be sure to enter the Code W and the $500.   That is what funds the HSA.  That $500 was totally tax free at the time of the deposit.  That $500 was not considered to be part of your taxable pay, and it reduced your income and did not get taxed for Social Security or Medicare.  The distribution should be expected to reverse that.

     I did a test case and found the entries to be bit tricky. Because you were not authorized to have an HSA, some entries will not make total sense. (I see what you mean with going in circles)  But as I see it, the $500 that is returned cannot qualify for any tax-free treatment because there was no real HSA. It appears that your employer simply returned your tax-free pay back to you. So that refund of pay should become taxable income that cancels the tax-free amount from the Code-W on your W-2.  As I see it, the objective is to get your income numbers as correct as possible. We will see that condition when we see the $500, which was refunded to you, listed on Line 21 of Schedule 1 as "other income" from the health savings account.
   Go to the "Less Common Income" section, and look for 1099-HSA and click to enter data.  Mark the entry as HSA.  You will need to indicate that something was paid from the HSA in order to get to the screen to enter data from your 1099-SA form.  Enter the 1099-SA data. Round to the nearest dollar and the nuisance 3 cents goes away. Be sure to record the Box 3 entry as the Code 2 excess distribution.  (This is good because the Code 2 will eliminate the 20% penalty.)  Click the "Yes" at the screen that asks if the HSA contribution was made by employer and excluded from Box 1 on your W-2. 
     It appears that all contributions were made via employer and captured with the Code W, therefore enter zero for your personal contributions.  Personal contributions are made with "after-tax" dollars, and that does not appear to be the case that you describe. I don't see any after-tax contributions described.  There appear to be no other contributions.  And click that you did not have Medicare.

    Here is where it got difficult because you might have had some insurance that does not make any sense within the context of HSA because you were not eligible to have the HSA. I believed it was proper to just check that you did not have HDHP coverage. Because if you indicate that you had some coverage, then the program goes down a path to calculate your tax-free components for your months of coverage. Your HSA contribution limit should have been zero. (You can do that by clicking "none" for all months. (Clicking any positive response for any month will result in a tax benefit that does not make sense to me. So I think "none" is correct response for the context of not having a real HSA.)  In the end the contribution limit should be zero if you had no HSA.  It will make the $500 taxable, which is what it should be to cancel the tax-free amount that is not appropriate. 

     Your description of events indicates that you did not overfund in any previous year. If the program asks if you “will make/have made” withdrawal of the $500, then I checked the I'll withdraw some of the excess by April 15."  When the program asks how much, I told it $500 has already been withdrawn . That is what your 1099-SA Code 2 is saying you did already.
     If the program complains, try going into the View/Forms section (to look at the forms view) and look at the Form 8889 to clear any pink boxes. I did that by selecting None for everything and self-only for line 1 in Part I.

     Note that the Line 14a of the Form 8889 has $500 and Line 14b also had $500 (be sure the Code 2 is entered on the 1099-SA form or that result will be different and wrong.). This lines 15 and 16 of the 8889 are zero and that avoids the penalty for the withdrawal.

     To see if the final tax return looks, OK, then print a preview copy. My Line 21 of Schedule 1 shows the $500 income along with the descriptions, “Form 8889 Health Savings Accounts 500.” That value sums into Line 22 of Schedule 1 and transfers over to the 1040 on Line 6. You should see that the $500 is effectively added back into your income to replace the $500 that was excluded from your W-2. If that is the case, then the income appears to be balanced. (You might have avoided a bit of Social Security/Medicare tax with those exchanges, but I don’t see any good or easy way to change that.) The important thing is that you should NOT be losing any penalty money with any entry on Line 62 of Schedule 4. My Schedule 4 is blank (and if blank, you won't even see it) due to the Code 2 entry from the 1099-SA.

    As far as what to do with the $500, it is you money and you get to decide what to do with it.  You might put it into this year's HSA, or perhaps you might open a Roth IRA.  I believe that the important thing is to control spending on stuff that is not needed and to begin saving for your future.  Good luck!