Here's my updated version:

<apex:page standardController="Room_Allocation__c"  extensions="Roommanager" tabStyle="Room_Availability_Search__tab">
    <apex:form >
    
        <apex:sectionHeader title="Room Availability Search"/>
                    
            <apex:pageBlock title="Search">
                <apex:pageBlockButtons >
                    <!-- Here's where {!Search} should be. -->
                    <apex:commandButton action="{!search}" value="Search"/>
                </apex:pageBlockButtons>
                <apex:pageBlockSection columns="2">
                    <!--
                        We are using a standard controller, so reference that controller instead.
                        We do this to make the code a bit more readable, as future coders won't have
                        to look at the controller to guess what the variable "room" is. Room is now
                        a private variable, too.
                    -->
                    <apex:inputField value="{!Room_Allocation__c.Search_Start_Date__c}"/>
                    <apex:inputField value="{!Room_Allocation__c.Search_End_Date__c}"/> 
                </apex:pageBlockSection>
            </apex:pageBlock>  
            
            <apex:pageBlock title="Available Rooms">
                <apex:pageBlockSection >
                    <apex:pageBlockTable value="{!availablerooms}" var="avail">
                        <apex:column value="{!avail.Lodge__c}"  />
                        <apex:column value="{!avail.Room_No__c}"  />
                        <apex:column value="{!avail.Name}"  />                        
                    </apex:pageBlockTable>
                </apex:pageBlockSection>   
            </apex:pageBlock> 
            
            <apex:pageBlock title="Booked Rooms">
                <apex:pageBlockSection >
                    <!--
                        By removing the global "room" variable, we avoid ambiguity between
                        this local variable and the global variable. This is to help
                        future developers that might look at the code.
                    -->
                    <apex:pageBlockTable value="{!rooms}" var="room">
                        <apex:column value="{!room.Room__r.Lodge__c}"  />
                        <apex:column value="{!room.Room__r.Room_No__c}"  />
                        <apex:column value="{!room.Room__c}"  />
                        <apex:column value="{!room.Name}"  />                        
                        <apex:column value="{!room.Start_Date__c}"  />
                        <apex:column value="{!room.End_Date__c}"  />
                        <apex:column value="{!room.Nights__c}"  />
                        <apex:column value="{!room.Reservation__c}"  />   
                        <apex:column value="{!room.Reservation__r.Reservation_Contact__c }"  />                         
                    </apex:pageBlockTable>
                </apex:pageBlockSection>   
            </apex:pageBlock> 
            
    </apex:form>
</apex:page>

 

public class Roommanager {
    public void search() {
        // Provide default values in the event that no dates are selected. This is optional.
        if(room.Search_Start_Date__c==null)
            room.Search_Start_Date__c=System.today().toStartOfMonth();
        if(room.Search_End_Date__c==null)
            room.Search_Start_Date__c=System.today().toEndOfMonth();
        
        // Swap the dates if start is before end. Optional nice logic.
        if(room.Search_Start_Date__c>room.Search_End_Date__c) {
            Date tempDate = room.Search_Start_Date__c;
            room.Search_Start_Date__c = room.Search_End_Date__c;
            room.Search_End_Date__c = tempDate;
        }
        
        // Remove previous search results, if any.
        rooms.clear();
        availableRooms.clear();
        
        // Find all the room allocations, add them to the list.
        rooms.addAll(
            [SELECT
                Id, Name, Start_Date__c, End_Date__c, Nights__c, Room__c, Room__r.Room_No__c, Room__r.Lodge__c, Reservation__c, Reservation__r.Reservation_Contact__c
            FROM
                Room_Allocation__c r     
            WHERE (
                (Start_Date__c >= :room.Search_Start_Date__c  AND Start_Date__c <= :room.Search_End_Date__c) OR
                (End_Date__c  >= :room.Search_Start_Date__c  AND End_Date__c  <= :room.Search_End_Date__c) OR
                (Start_Date__c <= :room.Search_Start_Date__c  AND End_Date__c  >= :room.Search_End_Date__c))  
            ORDER BY
                Room__r.Lodge__c ASC,Room__r.Room_No__c ASC, Room__r.Lodge__c ASC]);
        
        // Build a list of allocated rooms.
        Set<Id> allocatedRooms = new Set<Id>();
        for(Room_Allocation__c roomAlloc:rooms)
            allocatedRooms.add(roomAlloc.Room__c);
        
        // Add the rooms that are still available.
        // I like making queries indented for legibility.
        availableRooms.addAll(
            [SELECT
                Id, Name, Room_No__c, Lodge__c,
                (SELECT
                    Id, Name, Start_Date__c, End_Date__c, Room__c
                FROM
                    Room_Allocations__r)
            FROM
                Room__c
            WHERE
                Id NOT IN :allocatedRooms
            ORDER BY
                Lodge__c ASC, Room_No__c ASC]
        );
    }

    private Room_Allocation__c room;
    
    public List<Room_Allocation__c> rooms { get; set; }
    public List<Room__c> availableRooms { get; set; }
    
    // COnstructor. We initialize our memory variables here.
    public Roommanager(ApexPages.StandardController controller) {
        room=(Room_Allocation__c)controller.getRecord();
        rooms=new List<Room_Allocation__c>();
        availableRooms = new List<Room__c>();
        // Do a default search for our user.
        search();
    }
}

I added some optional features just as a suggestion, but note that the main problem that you had was that you were calling a null function {!room} instead of the {!search} function (see the VF page). Other than that, most of your code was sound. I assumed the queries were already working more or less okay and left them alone.

Note the other piece, where I referenced "room" by the controller instead of by a member variable in the VF page. Since they both point to the same place, you reduce (developer) confusion by using the standard controller when possible. There's no other real benefit here, but legible code is always a plus.

Finally, as you can see here, we consolidated the search logic into a single function, which provides the greatest flexibility.

Let me know how it works out.

Edit: Changed a private modifier to public.