Type.ForName is not working in apex

You are trying to pass a Batchable implementation where Schedulable is expected. This is just one reason why it is better to roll both into the same class. Regardless, your schedule method should pass in an instance of BatchEmails_Scheduler:

return system.schedule('<batch_name>', '<cron_string>', new BatchEmails_Scheduler());

Typically, rolling them together reduces the number of classes you have to maintain and adds little overhead:

public with sharing class MyBatch implements Database.Batchable<SObject>, Schedulable
{
    public void execute(SchedulableContext context)
    {
        Database.executeBatch(this);
    }
    public Database.QueryLocator start(Database.BatchableContext context)
    {
        // batch implementation
    }
    public void execute(Database.BatchableContext context, List<SObject> records)
    {
        // batch implementation
    }
    public void finish(Database.BatchableContext context)
    {
        // batch implementation
    }
}

For the first one, instead of:

affectedCasesByContactId.keyset().containsKey(...)

use either the Map method containsKey on the Map:

affectedCasesByContactId.containsKey(...)

or the Set method contains on the Set:

affectedCasesByContactId.keyset().contains(...)

For the second one instead of trying to access from the List:

trigger.new.get(...)

access from the Map:

trigger.newMap.get(...)

Take a look at the available Trigger Context Variables and also do some Googling about lists and maps to get clear about how to use those most effectively.