The problem was that you were querying opportunities against opportunity contact role IDs, so there would be no results.

Regardless, this code is horribly inefficient, in any case, using 12 queries instead of 2 (NOTE: if you know what Agent_Status__c is, you could make it just 1 query), and 4 DML statements instead of 1.

Here's a revised version of your code:

public class UpdateOpportunityStageOnContact {

        public static void agentstatus(Contact[] contactsToUpdate) {
            Map<Id, Contact> contacts = new Map<Id, Contact>(
                [SELECT Id, Agent_Status__c
                 FROM   Contact
                 WHERE  Id IN :contactsToUpdate AND
                 Don_t_Update_Opportunity_Via_Flow__c = FALSE AND
                 Agent_Status__c IN ('SureLC Packet Complete','Carrier Contracting Complete','First Case Submitted','Client')]);
            Opportunity[] opps = new Opportunity[0];
            OpportunityContactRole[] roles = [SELECT Id, ContactId, OpportunityId FROM OpportunityContactRole WHERE ContactId IN :contacts];

            Map<String, String> convMap = new Map<String, String> {
                'SureLC Packet Complete' => 'SureLC Packet Complete',
                'Carrier Contracting Complete' => 'Carrier Contracting Complete',
                'First Case Submitted' => 'First Case Submitted',
                'Client' => 'Closed Won'
            };

            for(OpportunityContactRole contactRole: roles) {
                opps.add(new Opportunity(Id=contactRole.OpportunityId, StageName=convMap.get(contacts.get(contactRole.ContactId).Agent_Status__c)));
            }

            update opps;
        }
}

Assuming that contactsToUpdate is from a trigger, you can just loop through the records, and save another query:

Map<Id, Contact> contacts = new Map<Id, Contact>();
Set<String> statusValues = new Set<String> { 'SureLC Packet Complete','Carrier Contracting Complete','First Case Submitted','Client' };
for(Contact record:contactsToUpdate) {
    if(!record.Don_t_Update_Opportunity_Via_Flow__c && statusValues.containsKey(record.Agent_Status__c)) {
        contacts.put(record.Id, record);
    }
}

A total of 1 query to get all the data you need to update the opportunities, and 1 DML statement to update all the opportunities involved.

You're correct on two of three points. ContainsKey should have been contains, and :contacts should have been :contacts.keySet(). However, the == FALSE is not necessary. The point of a condition is to have a true or false value. Since a checkbox is already a true or false value, we don't need to compare it against true or false. The "!" at the beginning of the condition inverts the flag, such that true becomes false and false becomes true. In other words:

Boolean falseValue = !true;
Boolean trueValue = !false;

if(trueValue) {
    System.Debug('This is true!');
}
if(!falseValue) {
    System.Debug('This is also true!');
}

Here's the final revision you should need:

public class UpdateOpportunityStageOnContact {

        public static void agentstatus(Contact[] contactsToUpdate) {
            Map<Id, Contact> contacts = new Map<Id, Contact>(
                [SELECT Id, Agent_Status__c
                 FROM   Contact
                 WHERE  Id IN :contactsToUpdate AND
                 Don_t_Update_Opportunity_Via_Flow__c = FALSE AND
                 Agent_Status__c IN ('SureLC Packet Complete','Carrier Contracting Complete','First Case Submitted','Client')]);
            Opportunity[] opps = new Opportunity[0];
            OpportunityContactRole[] roles = [SELECT Id, ContactId, OpportunityId FROM OpportunityContactRole WHERE ContactId IN :contacts.keySet()];

            Map<String, String> convMap = new Map<String, String> {
                'SureLC Packet Complete' => 'SureLC Packet Complete',
                'Carrier Contracting Complete' => 'Carrier Contracting Complete',
                'First Case Submitted' => 'First Case Submitted',
                'Client' => 'Closed Won'
            };

            for(OpportunityContactRole contactRole: roles) {
                opps.add(new Opportunity(Id=contactRole.OpportunityId, StageName=convMap.get(contacts.get(contactRole.ContactId).Agent_Status__c)));
            }

            update opps;
        }
}

Map<Id, Contact> contacts = new Map<Id, Contact>();
Set<String> statusValues = new Set<String> { 'SureLC Packet Complete','Carrier Contracting Complete','First Case Submitted','Client' };
for(Contact record:contactsToUpdate) {
    if(!record.Don_t_Update_Opportunity_Via_Flow__c && statusValues.contains(record.Agent_Status__c)) {
        contacts.put(record.Id, record);
    }
}