Because I filed electronically, there is no signature on my 1040. Do government agencies accept this "unsigned" 1040 as "official" or do I need one with a signature?

You need to give someone a copy?  Print it out and sign it.

There is also a space in the 1040 for "Declaration and Signature of Electronic Return Originator". Does this need to be signed?

and if so, by whom?

No it doesn't ... TT is not a paid preparer ... the program should have entered "self prepared" in that section.

"Declaration and Signature of Electronic Return Originator"   doesn't show up on the form 1040 ... <a rel="nofollow" target="_blank" href="https://www.irs.gov/pub/irs-pdf/f1040.pdf">https://www.irs.gov/pub/irs-pdf/f1040.pdf</a>  .... that would be on a form 8879 which is only used by paid preparers ... <a rel="nofollow" target="_blank" href="https://www.irs.gov/pub/irs-pdf/f8879.pdf">https://www.irs.gov/pub/irs-pdf/f8879.pdf</a>

When you efile you sign it electronically ...  usually where the signature is the efile PIN would be listed however do to security reasons  the program doesn't put it there.  

If you need to give someone a signed copy then print it and ink sign it manually.