Q:

Difficult time understanding 1040-ES and Late Payments Penalties

I've spent the entire day trying to figure out how 1040-ES's work.  
If you make quarterly payments of Estimated Taxes, do you also file the regular 1040 on April of following year?  Turbo Tax states that 1040-ES's are for the current year, while the 1040 is for the previous year, so does that mean that if you file quarterly 1040-ES Vouchers + Payments in 2017, then in 2018 will you also file a 1040-ES in January of 2018 as well as a 1040 for the prior year of 2017? 
I've missed 1040-ES payments for the past two quarters, is there a way to make-up the payments in the next 2 remaining quarters either by overpaying in that quarter what is due in taxes or do I have to wait until the end of the year?  How do you pay a late payment for a prior quarter and not have it be applied to the current quarter - is this done with the vouchers stating which quarter the payments are for? 
So, for future reference, Safe Harbor IRS code requires that I pay either 100% of the previous year's taxable amount for the current year - in order to avoid penalties from fluctuating income (either from a windfall or from shortfall)?  If I make less than expected the following year, do I need to file a regular 1040 to get a refund or will the IRS automatically refund me? 
I became n independent contractor and made 15k last year (my only source of income) and ended up paying 2k in taxes after deductions - I had planned to find find a full-time job this year and use the taxes withheld to cover my freelance work since I really didn't make enough to survive in Self Employment.  
This year however, I had an unexpected windfall for a full time contract job that kept extending, and I plan to make close to 100k.  I'm worried about the taxes I'll owe the next year since I missed the first two quarters. 
What can I do to minimize my Tax Penalty?  I would like to make tax payments now instead of waiting for the end of the year.  I'm hoping the IRS will apply the penalty up to the overpayment on the remaining quarters but the IRS's Publication 505 stated that even if you're due a refund at the end of the year for overpayment, you could still owe a penalty?  Would they just keep the refund and apply it towards the penalty? 
Thanks! 

A:

The answers to most of your question are in IRS Publication 505, from which my answers are adapted. 

If you make quarterly payments of Estimated Taxes, do you also file the regular 1040 on April of following year?  Turbo Tax states that 1040-ES's are for the current year, while the 1040 is for the previous year, so does that mean that if you file quarterly 1040-ES Vouchers + Payments in 2017, then in 2018 will you also file a 1040-ES in January of 2018 as well as a 1040 for the prior year of 2017? 

You do not really have to "file" 1040-ES at all.   If you pay by check or money order, you must mail the small voucher form with your payment, but this basically just contains your identifying info and the amount you're paying; there's no calculation on the form.   If you pay electronically (e.g., via EFTPS), which is much easier, you don't need to mail or file anything. 
Regardless, you do indeed still have to file the regular 1040 the following year as usual.   The estimated tax payments are just payments; it is the form 1040 that contains the statement of what your income was and the computation of how much tax you owe.   On that 1040 there is a line (line 65 for the regular 1040 last year) on which you report how much you have already paid via estimated tax payments. 

I've missed 1040-ES payments for the past two quarters, is there a way to make-up the payments in the next 2 remaining quarters either by overpaying in that quarter what is due in taxes or do I have to wait until the end of the year?  How do you pay a late payment for a prior quarter and not have it be applied to the current quarter - is this done with the vouchers stating which quarter the payments are for? 

Once the deadline has passed, you cannot make a payment for the missed quarter.   You can "make up" your missed payments in the sense that you can still pay all the tax you owe, but you may still face a penalty for not paying it soon enough.   As described in IRS Publication 505:

Because the penalty is figured separately for each payment period, you may owe a penalty for an earlier payment period even if you later paid enough to make up the underpayment.  

Basically what this means is that the penalty for not paying in a particular period is separate from the "global" penalty for not paying enough overall during the year.   As a simple example, suppose that you estimate you will owe $4000 in taxes, and suppose your estimate is correct (i.e., at the end of the year once you look at how much you actually made, it turns out your tax liability is indeed $4000).   You would ideally pay this in four payments of $1000 each.   If you forget the first two payments, you could make the last two $2000 each.   In this case, you may be penalized for not making the first two payments on time, but at least you did pay all your taxes for the year.   If instead you just pay $1000 for each of the latter two payments, you will only wind up paying $2000 altogether, so in addition to the quarterly penalties, you may face a "global" penalty for not paying enough altogether.   This means it's a good idea to make up missed payments if you can, but it can't completely save you if you did indeed owe taxes for the missed periods. 

What can I do to minimize my Tax Penalty?  I would like to make tax payments now instead of waiting for the end of the year.  I'm hoping the IRS will apply the penalty up to the overpayment on the remaining quarters but the IRS's Publication 505 stated that even if you're due a refund at the end of the year for overpayment, you could still owe a penalty?  Would they just keep the refund and apply it towards the penalty? 

As mentioned above, you definitely should make payments for the remaining quarters if you expect to have tax liability for the year (which it sounds like you do). 
Since the penalties are figured when you file form 1040, you can't directly pay them ahead of time.   There's no way to use your third payment to "pay off" the penalty for not making the first payment.   However, Publication 505, in the section about Form 2210, indicates that in figuring the penalty, you subtract any overpayment from the penalty, so in this sense, if you are due a refund, it will automatically be "applied" to the penalty. 

This year however, I had an unexpected windfall for a full time contract job that kept extending, and I plan to make close to 100k.  I'm worried about the taxes I'll owe the next year since I missed the first two quarters. 

This is the most complicated part of your question and is actually too complicated to be fully answered here.   Suffice it to say that there are two quite different possibilities:

A. You began making money at the beginning of 2017, on pace to make $100k by the end of the year.   This was "unexpected" in the sense that it's a lot more than you made last year, but it began at the beginning of the year. 
B. You made little or no money for the first half of 2017, and then unexpectedly began making a lot of money. 

In situation A you will probably face a penalty, since you were making enough money to know you were going to owe taxes, and you didn't pay them.   There is nothing you can do except increase your last two payments to cover your total liability (along the lines in the example above about making two $2000 payments instead of four $1000 payments). 
In situation B you may not face a penalty at all, since you earned so little during the first two periods that you could legitimately expect that you wouldn't owe taxes at the end of the year. 
To answer this part of your question, use Form 2210, specifically the "Annualized Income Installment Method".   This form is a bit of a doozy, but essentially what it does is look at how much you actually earned in each part of the year.   If your earnings were widely different during different parts of the year, you may reduce or eliminate your penalty by using this method.   You'll have to go through the form yourself with your specific numbers.