[SalesForce] Attribute name “checked” associated with an element type “input” must be followed by the ‘ = ‘ character

You need to define the posType with getter/setter and set the default value in your constructor, the rest is correct:

public String posType { get; set; }

public YourClass(){
    posType = 'Consultant';
}

public List<SelectOption> getItems() {
    List<SelectOption> options = new List<SelectOption>();
    options.add(new SelectOption('Mentor','Mentor'));
    options.add(new SelectOption('Consultant','Consultant'));
    options.add(new SelectOption('Permanent','Permanent')); 
    return options;
}

Page:

<apex:selectRadio value="{!posType}" layout="lineDirection">
    <apex:selectOptions value="{!items}"/>
</apex:selectRadio>

The code above works on my dev org.

Can you just use outputText instead if it needs to be disabled? Not the same appearance but might work for you?

<apex:outputText id="impressions" value="{!drop.impressionsRequested}" rendered="{!dropObj.impressionsDisabled}"/>

<apex:inputText id="impressions" value="{!drop.impressionsRequested}" rendered="{!NOT(dropObj.impressionsDisabled)}"/>

Or instead out output text you could just use a disabled HTML input element so it would maintain the same look:

<apex:outputPanel layout="none" rendered="{!dropObj.impressionsDisabled}">
    <input type="text" value="{!drop.impressionsRequested}"/>
</apex:outputPanel>