1. It is not (normally) necessary for your 2019 tax return to have her previous returns in hand since individual tax returns are largely independent of each other from year to year. However, for your peace of mind, I would encourage to continue to try to get the transcripts. Of course, it is possible that you can't get them because the returns were never filed.
In any case, file your 2019 return in any case even as you are searching for other things.
2. It appears to me that the IRS will allow her SS number to be used as her identification number, even with the caveat "Use for Work Authorization Only". At least, that is the conclusion that I draw from this IRS page. The second image wouldn't make any sense if the taxpayer were not able to use this Social Security number.
3. I don't see any immediate problem to you e-filing for 2019; however, you are correct to assume that even if the e-file failed for some reason, then you can print the return using TurboTax and then mail it yourself.
A typical reason that e-filing is rejected is because the name of the taxpayer on the tax return does not match the name of the taxpayer on the SS card with the same SS number. This is often a difficulty for newly-weds (because of the name change) and for those who do not have the normal American paradigm for a name (first name, last name). Hispanics in particular have difficulties because they often use both the mother's and the father's last name, so there exists the chance for confusion in that the taxpayer thinks of one name as the "last name" while the SSA thinks of the other name as the "last name".
Just use whatever name is on the SS card...and then if you need to change the name on the card, do it after your e-file is successful this year.
4. Without more information, it is not even clear to me that your wife would have been required to file a tax return prior to 2019. What you might do is if the IRS can't show you her W-2s from those years (NOTE: there is more than one type of transcript so make sure you ask for a transcript that shows income items like W-2s) is ask the University instead. Once you find any paperwork, then you can go through the separate process of determining whether or not she was required to file (this may require the services of a local tax pro who is familiar with filing returns for foreign taxpayers here on a visa).
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