Re the Section 121 exclusion, it may (or may not be) a partial exclusion since Mary appears to have lived there until October of 2022. Mary inherited the property in late December of 2020, so ownership would be (slightly) less than 2 years. More details are needed for that purpose.
Re Peter, the transaction, as stated, would appear to be a part gift/part sale.
If, for example, Mary pays Peter $125,000 for that 1/3 share when the FMV is $175,000, then Peter would technically have a loss against his basis of $155,000 (which loss would not be recognized due to related party rules per Section 267).
Peter would also have made a gift to Mary of the difference between the FMV and Peter's basis (which would be $50,000 and reportable, obviously). Mary's basis would then be the greater of Peter's basis or what she paid for Peter's 1/3 share (so Mary's basis for Peter's 1/3, per Section 1.1015-4, would be $155,000).
