To answer your questions in order:
1 - Having zero Federal tax withheld can lead to underpayment penalties, but since you are planning to file a joint return and your job withheld Federal income tax, then it is less likely to create an underpayment situation. Instead, your tax withholdings reported on your W-2 will help to 'pay' the income tax for the joint return. The underpayment penalty on a joint return considered the total income and withholdings, not each person's share being reported individually.
2 - E-filing should not be prohibited when filing a Married Filing Joint tax return where there is a W-2 with zero withholdings included in the return. The rules may be different if filing as a Registered Domestic Partnership (RDP).
3 - The only reason you would need to request a corrected W-2 would be if you can compare your spouse's last pay stubs from the jobs and they showed that Federal taxes were withheld. If the pay stubs show withholdings, but the W-2s do not, then the W-2 need to be corrected. Employers may be able to make the corrections quickly, but if not, you can always file an amended return later if you do not want to wait for the correction.
4 - There may not be anything that you need to do. In the future, be sure that any W-4 that is turned in at the beginning of employment shows the proper filing status and exemptions. If too many exemptions are claimed, then the chance of no taxes being withheld is greater. People sometimes choose to do this to increase the money in pocket when they get paid.
