[SalesForce] Update a picklist value with a text field’s value using workflow

How about trigger compare? You must create 2 triggers make it work.

on object 2 you must have after insert/update (and before delete but I will skip it) actions in method:

Set<String> matchedFields = new Set<String>();
object_1__c[] obj1List = new object_1__c[]{};

for(object_2__c obj : Trigger.new){
    matchedFields.add(sds__c);
}

for(object_1__c obj : [SELECT Id, sds_chek__c
                       FROM object_1__c
                       WHERE sds__c IN : matchedFields]){

   if(obj.sds_chek__c == 'No'){
       obj.sds_chek__c = 'Yes';
       obj1List.add(obj)
   }
}

update obj1List;

You should also have trigger on before insert object_1__c that will check if in system already is some matching record

If you have any other question please let me know

I have changed the below controller. I initially didn't see that you're not calling any method from the apex:actionSupport tag. I have changed the below code (Apex / VF) and added some comments.

public with sharing class DynamicCasesExtension {

    public Case c { get; set; }
    public String objectName { get; set; }
    public List<String> objectFields { get; set; }
    public List<Case> SObjectListToShow { get; set; }

    public DynamicCasesExtension(){
        objectName = 'Case';
        //set the status to the value you need. In your case I assume it is coming in as a parameter. This is to be set only for the first load. 
        c.Status = ApexPages.currentPage().getParameters().get('c.Status');
        //SObjectListToShow = getSObjectList();
        generateList();
    }

    // a new method which will be called everytime the value in the picklist is changed
    public void generateList() {
        SObjectListToShow = caseSObjectList();
    }

    // Getting field list for the sObject. Changed the name of the method.
    public List<Case> caseSObjectList(){
        .
        .
        .
        //Change it to c.Status so that it gets set to the value selected from the page.
        String caseStatus = c.Status;
        .
        .
        .
    }

}

VF Page:

<apex:actionSupport event="onchange" rerender="result" action="{!generateList}"/>