The file formats are the same, so no issues there. I have a similar version discrepancy between my work computer and home laptop, and work on documents on both without any issues.
The laptop doesn't have the Times New Roman font installed, which is why you're not seeing it in LibreOffice. I believe this is due to licensing issues.
I don't think there will be problems with Sigil, but I always remove any reference to specific fonts in my ebooks. I prefer to let the reader use whatever font they've selected on their ereader.
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