To answer both questions:
Yes, I sent the contract to a test email and I was able to edit the fields as a signer.
See attached.
Using your test contract file I could reproduced the issue.
And the reason for the issue is simple, Adobe Acrobat Sign (at least if used via the Acrobat "Request e-signatures" tool) flattens existing digital signatures.
The field locking mechanism you employ essentially adds the information to the AcroForm signature field that once it is signed, the fields with the given names shall be considered locked.
But the Request e-signature / Acrobat Sign workflow starts by flattening that AcroForm signature field, i.e. it keeps just the appearance and throws away everything else, in particular the field locking information.
You can see that your signature (except the appearance) have been removed by clicking its visualization on the page. In your original file the Signature Validation Status dialog pops up. In the file versions you retrieved back from Adobe Sign it doesn't. Also you can see on the Acrobat Signatures panel that your signature now is missing and only a certification signature from Adobe Acrobat Sign is there.
Thus, the fields in the form shown to the client can be freely edited.
So when using Adobe Acrobat Sign, make sure that the PDF only contains fields for data the client shall be able to change. Furthermore, also consider including yourself in the Acrobat Sign signers instead of signing locally - as soon as you request e-signature, your former digital signatures are removed and only their visual representation remains and a historical remark on the audit report.