[SalesForce] Using REGEX in Visualforce

You will want to use dynamic soql because then you can choose which things to query for depending if you user selects them. Try this.

String displayContactsStr = 'select id, Name, Domain__c, City__c, FirstName, LastName, Email, Country__c from Contact where Name <> null '; 
if (contactData.Domain__c != '') {
   displayContactsStr += ' and Domain__c = \'' + contactData.Domain__c +  '\'';
}
if (contactData.Country__c != '') {
   displayContactsStr += ' and Country__c = \'' + contactData.Country__c +  '\'';
}
if (contactData.City__c != '') {
   displayContactsStr += ' and City__c = \'' + contactData.City__c +  '\'';
}
displayContacts = (List<Contact>)Database.query(displayContactsStr);

rendered="{!output}" evaluates boolean expression.

In your code output is String type. Change it to Boolean like this.

public Boolean output {get;set;} 

// First Condition: When MRN is blank 
if( !string.isBlank(tr.mrn))            
{
    if (RowList.size() > 1){
        ApexPages.addMessage(
        New ApexPages.Message(
        ApexPages.Severity.FATAL, 'Multiple records found. Please try with MRN'
        ));
        return null;                       

      }
      output = false;
}
else
{
      // Second Condition: When MRN is Not blank
    output = true; 
} 

Otherwise, use rendered="{!output=='true'}"

Update

According to your logic, data will be displayed in following conditions:

When output is true, string.isBlank(tr.mrn) is blank and RowList.size() > 1

So, I think you have wrongly written the logic. According to the comment in the code, the correct logic will be as follows:

// First Condition: When MRN is blank 

if(string.isBlank(tr.mrn))            
{
    if (RowList.size() > 1){
        ApexPages.addMessage(
        New ApexPages.Message(
        ApexPages.Severity.FATAL, 'Multiple records found. Please try with MRN'
        ));
        return null;                       

      }
      output = false;
}
else
{
    // Second Condition: When MRN is Not blank
    output = true; 
}