How can I check the status of a mailed 1040x? It was sent in July of 2020 but I cannot find it in the IRS system?

   The IRS has not been working on mailed returns until fairly recently---and an amended return could only be filed by mailing it in until late August 2020 when the IRS began to accept e-filed 2019 amended returns.   Under normal circumstances you would be able to see something about your mailed in Form 1040X after about three weeks, but that is not the case right now.  The usual 4 months for processing an amended return is going to take longer.  Sorry.

https://www.cnn.com/2020/04/20/politics/irs-mail-trailers-covid/index.html

MAILED RETURNS

Per the IRS website IRS Operations During COVID-19: Mission-critical functions continue:  

IRS operational status and alternatives

Processing Delays for Paper Tax Returns: Taxpayers should file electronically through their tax preparer, tax software provider, or IRS Free File. 

We’re experiencing delays in processing paper tax returns due to limited staffing. If you already filed a paper return, we will process it in the order we received it.

Do not file a second tax return or contact the IRS about the status of your return.

 If you mail a tax return (or a payment) to the IRS, it is a good idea to use a mailing service that will track it like UPS or certified mail so you will know it was received.

 TurboTax will not know anything about your amended return, and nor will TT know that you put your 1040X  in an envelope and took it to a mailbox.  TurboTax does not get updates on mailed  (or e-filed) returns.

When the IRS finally opens its mail and begins to process mailed returns you can check the status on the IRS site.

https://www.irs.gov/Filing/Individuals/Amended-Returns-(Form-1040-X)/Wheres-My-Amended-Return-1

https://www.irs.gov/filing/individuals/amended-returns-form-1040-x/amended-return-frequently-asked-q...

As fir your hopes to receive a stimulus check:

The stimulus check is an advance on a credit you can receive on your 2020 tax return.  If something went wrong or you did not get the stimulus check this year, you can get it when you file your 2020 return in early 2021—if you are eligible.It will end up on line 30 of your 2020 Form 1040.

https://www.irs.gov/newsroom/economic-impact-payment-information-center-topic-j-reconciling-on-your-2020-tax-return

https://www.irs.gov/coronavirus/coronavirus-tax-relief-and-economic-impact-payments-for-individuals-and-families