@TaxStyle You stated the facts below.  I'm NOT going to go off on a tangent to discuss things like corporate formalities, commingling corporate funds, whether the biz was run like a corporation, what "setting up a company" could also mean, or speculations like whether the owner pocketed the funds or she filed a schedule C or whether she suffers from chronic constipation....  Here, I generally take the facts literally and address the questions as asked based on the given facts.  If other tax angles are there to be considered along the line of my discussion, I will mention them.

Your questions were very clear to me.  So here it goes. 

Stated facts:  So later YOU found out it is an C corp when YOU prepared her 2020 return, and she is the sole owner and the only member of the C corp. However, her 2019 return (initial return) was filed using S corp form 1120-S.  Now she received a letter from IRS saying that her 2019 return cannot be processed, because it doesn't match.

Basically a tax can of worms, or is it a can of tax worms.  I have NEVER had worms like these.  Below, I'll tell you how I would approach it - NOT answers.  Just suggested directions. 

Actually, part of what I suggest below is an expansion of what @TaxGuyBill said.  He nailed it on the head.

Take a step back.

What type of taxation do YOU think is best for her situation?  S-corporation?  C-corporation?  Sole Proprietor?  Something else?

As your client's advisor, you may want to help her to decide the best for her business.  That said, assuming the incorporation was NOT voided, even if it was decided an entity OTHER THAN  a corp. was the best, practically, you still have to deal with the corp filing UNTIL it's dissolved.

SO, for years 2019, 2020 and 2021 (and soon 2022 unless the corporation is dissolved before 12/31/2022), you still have to deal with either 1120 or 1120S filings.

*** Remember when you read on.  I'm just suggesting directions for your research.  I don't know the feasibility  ***

2.  Plan A - If client decided to go with S corporation.

2.1  Look into a late F2553 filing.  See Rev Proc 2017-34.  the corporation is still within the 3 year and 75 days period [per Section 4.02(2)] .  The way to do it (but see 2.2 below) would be to file (refile the original) F1120S with Form 2553, both bear the "FILED PURSUANT TO REV PROC...." marking - that is, on both F1120S and F2553.  

2.2  Amend 2020 F1120 to all zero.  File it as a final return.

2.3.  File 2020 F1120S with F2553.  Include the explanation as part of F2553.

2.4.  NO need to amend 2019 1040/540.  

2.5.  Late filing penalties for 2019 and 2020 F1120S.  (See 4 below)

2.6.  In particular, you need to research: "Would the filing of 2020 F1120 preclude a late F2553 filing??"  You're totally on your own on this.  I have ZERO clue.  If YES for the question, Scratch Plan A.  

2.7.  @TaxGuyBill  said: But I think 2020 has already been filed as a "C", it would mess up the ability to file the late S-election.  Source?

3.  Plan B - If client decided to go with C corporation OR if 2.6 above is YES.

3.1.  File F1120 for 2019, which would be hit with Late filing penalties.  (You stated net income for 2019). 

3.2.  The F1120S was rejected.  NO action needed for the rejected F1120S.

3.3.  Amend  1040/540 to back out the K-1 income.

3.4.  NO change for 2020 filing.  (F1120 already filed)

4.   Late penalties

4.1. See 3.1 and 2.5 above for the filings subject to penalties.

4.2.  Either pay the penalties OR write to request for penalty waiver.  

4.3.  For penalty waiver, research IRM 20.1 - Particularly 20.1.1.3.6 and 20.1.1.3.3.2.1 to see if the First offense waiver is still available and applicable.  

https://www.irs.gov/irm/part20/irm_20-001-001r#idm1397614523489123

5.  Your questions.  

5.1  I think the above would either answer your Q1 and Q2 or give you enough headache about them

5.2   Your Q3:  is there is a way to just ask IRS to apply the refund to C Corp tax, and she doesn't need to do anything from her end?  NOPE.  These days you would have a hard time moving the credit for the tax payments made for one year to another for the same entity/tax form.  NO WAY to apply refund for 1040 to 1120...

5.3.  Your Q4 4.How does the state return(California) gonna get  affected?  since there is no letter/notice received from State. Do I wait for everything from federal level is cleared then work on state returns or I can do it at the same time? 

CA recognizes Federal S election.  NO separation action needed.  It makes sense to do it at the same time.

5.4  Your Q5   I am assuming attached a letter to explain to the state authority about the situation is needed instead just sending in new/amending state returns?

Who is there to read your letter?  Tax filing matters are basically "Form driven".  If there is NO form, how can FTB take action?  I used to work for the state (NOT FTB).  For received letters that we didnt know what to do, we just put in another guy's in box.  As such, your fine letter will just go from in boxes to in boxes, so to speak.

Hope this helps.

(Remember, I'm just suggesting the directions or giving you my take.  My points are NOT the final answers.)