Do I need to enter the name of the settlement company who handled the closing of my investment property which I inherited?

If a HUD-1 exists and has a person identified as the agent, or there is a settlement agent otherwise identified, then they are listed as the "Filer" in the first box of the 1099-S.  If a HUD-1 wasn't used, or no settlement agent is designated, then whoever prepared the 1099-S is the "Filer" in the first box.  Presumably you are the "Transferor" (seller), so your SSN is listed separately from the Filer's SSN.

See the section titled "Who Must File" on page 2 of the instructions for Form 1099-S at:
   https://www.irs.gov/pub/irs-pdf/i1099s.pdf

Please answer the question asked:  Where do I enter (into TT) the name of the Settlement Company?

It's similar to a 1099-INT.  You enter the name of the institution that paid you interest so the IRS can match your return to the 1099-INT.