I apologize again for the late response. Please view my answers below:

1. I re-written the code more generically so that you can replace the API values with the values from your org. Please review the code below for reference: 

In Developer Console, go to File, hover over new and select Apex Class. Name the Apex Class, "LeadTriggerHelper" and click Ok. Afterwards, copy and paste this code:
 

public class LeadTriggerHelper {

    public static void updateCompanyName(List <Lead> leadList) {
        
        //Store all look up object records obtained via SOQL into a List.
        List <API_of_LookUpObject> lookUpObjList = [SELECT API_FIELDS FROM API_of_LookUpObject];

        //Replace All the API References in the statements above and below with the APIs 
        //from your org.
        //Replace API_FIELDS with all the fields you want returned in the query with a comma 
        //in-between each of them
        //Example: [SELECT Id, Name FROM Account]; 
        
//Create a Map to store the ID and Look Up Objects. Pass the list created above into it.
Map <Id, API_of_LookUpObject> lookUpObjMap = new Map <ID, API_of_LookUpObject>(lookUpObjList);
        
        //Iterate through Leadlist using for Loop
        for (Lead singLd : leadlist) {
            
            if (singLd.compName__c == null) {
                
                singLd.adderror('You must select a Look Up Record');

            }
                else if (singLd.compName__c != null) {    
                     
                singLd.Company = lookUpObjMap.get(singLd.compName__c).API_FIELD;
/* 1 - Replace API_FIELD in the statement above with the field you want from the Lookup Object.

    2 - Please note that the data type for this field has to be either a string or ID or else an error will occur.  
  
   3 - Also make sure that this API Name is also stated in the SOQL statement mentioned above after the word SELECT or else an error will occur
  */
            }
        } 
    }
}

After you make the API changes and save it, go to Developer Console, go to File and Hover over new and select Apex Trigger. Name it LeadTrigger and select Lead as the sObject.

Copy and paste the code below:

trigger LeadTrigger on Lead (before insert, before update) {

    LeadTriggerHelper.updateCompanyName(Trigger.new);
    
}

As a best practice, one should always put the code of a trigger within a separate Apex Class and just reference the method in the Apex Trigger. The code will be easier to manage just in case you decide to add more trigger methods in the future.

2 - I am not sure what you are referring to when you state user types. Lets say a user selects one of the look up records in the Look up field. Would you like for the standard Company field to state "Existing Company" rather than having it be replaced with the name of the custom company record? If that is what you meant, then yes that is possible. Minor updates would have to be made to the code.

3 - Only updates and insertions of a new or existing lead record will cause this trigger to execute. Thus, if the "+" is shown on lead records and a user clicks it to perform some type of change on the record, then this trigger will execute. However, if the "+" is shown only on the custom object, then it should not trigger a change to the lead records when used. To better understand, is the "+" a button? and where does it usually appear? A sample screenshot would be helpful to get a better understanding.

4 - That is correct. If a user does not have editing access, then they should not be able to utilize the "+" sign. Profiles of those you want to restrict access to should be read only. Use permission sets to give certain individuals access to it.

If you have any more questions or concerns, please feel free to reach out to me. I'm am more than happy to help!