The “Anonymous Household” behavior can occur due to Salesforce NPSP behavior when the last Contact associated with a Household Account is removed during a Contact merge. To avoid this, merge the Household Accounts first, followed by the associated Contacts.
If Contact names are duplicated after a merge, such as Tom Smith → Tom Thomas Smith Smith, check the Combine Fields setting under Master & Field Rules in the Dedupe Action. If Combine Fields is enabled for First Name and Last Name, values from both records can be concatenated during the merge. Disable Combine Fields for these fields and retry the merge.