TSQL Fill gaps in months and repeat values in column

Question 1

It's not clear how USERNAME populated. I assume you have same USERNAME on project here. CTE below just build a DATE table, if you have your own date table, you can skip this part.

SQL Fiddle

DECLARE @ENDDate DATETIME

SELECT @ENDDate =  MAX(END_DATE) FROM tblRawData

;WITH tblDate AS 
( SELECT CAST(MIN(START_DATE) AS DATE) AS [date]
  FROM tblRawData 
  UNION ALL
  SELECT DATEADD(month,1,[DATE])
  FROM tblDate
  WHERE [DATE] < @ENDDate
 )
SELECT
  d.[date]
  ,r.[Project]
  ,UserName =   (SELECT MAX(USERNAME) FROM tblStage1 ts WHERE r.PROJECT = ts.PROJECT)
  ,Percentage = (SELECT ISNULL(MAX(Percentage),0) FROM tblStage1 ts WHERE r.PROJECT = ts.PROJECT AND ts.[date] <= d.[date])
FROM tblDate d
INNER JOIN tblRawData r
  ON d.[date] between r.[START_DATE] AND r.[END_DATE]
ORDER BY 2,1
OPTION (Maxrecursion 0)

EDIT: Just found out the date is increased by month. I update the CTE query. However, you need make sure you have all project start and end date at first day of the month.

EDIT Base on your new sample date. the query became a little ugly now, however, it works. I can not think better solution right now.

New SQL Fiddler

DECLARE @ENDDate DATETIME

SELECT @ENDDate =  MAX(END_DATE) FROM tblRawData

;WITH tblDate AS 
( SELECT CAST(MIN(START_DATE) AS DATE) AS [date]
  FROM tblRawData 
  UNION ALL
  SELECT DATEADD(month,1,[DATE])
  FROM tblDate
  WHERE [DATE] < @ENDDate
 )
,ProjectList AS (
 SELECT Project,UserName
 FROM tblStage1
 GROUP BY Project,UserName
)
,cte AS (
SELECT
  d.[date]
  ,r.[Project]
  ,UserName = pl.Username
  ,CloseDate = (SELECT MAX(ts.[date]) FROM tblStage1 ts WHERE r.PROJECT = ts.PROJECT AND ts.UserName = pl.UserName AND ts.[date] <= d.[date])
FROM tblDate d
INNER JOIN tblRawData r
  ON d.[date] between r.[START_DATE] AND r.[END_DATE]
CROSS APPLY ProjectList pl
)
SELECT cte.[date],cte.project,cte.UserName,ISNULL(t.[PERCENTAGE],0) AS PERCENTAGE
FROM cte
LEFT JOIN tblStage1 t
 ON cte.PROJECT = t.PROJECT 
  AND cte.UserName = t.UserName  
  AND cte.CloseDate = t.[Date]
ORDER BY 2,3,1

Question 2

You could do this with the APPLY operator as well...

WITH DateList (Project, MonthStart) AS
    (
        SELECT 
            project
            , Start_Date
        FROM 
            tblRawData
        UNION ALL
        SELECT 
            dl.Project
            , DATEADD(MONTH, 1, dl.MonthStart)
        FROM 
            DateList dl
        JOIN
            tblRawData r 
            ON 
                r.Project = dl.Project
                AND 
                dl.MonthStart < r.End_Date
    )
SELECT 
    dl.Project
    , lastUser.UserName 
    , dl.monthstart [Date]
    , ISNULL(pct.Percentage, 0) Percentage
FROM 
    DateList dl
CROSS APPLY
    (
        SELECT 
            TOP 1 
            USERNAME
        FROM
            tblStage1 t1
        WHERE 
            t1.Project = dl.Project
        ORDER BY 
            t1.Date DESC
    ) lastUser
OUTER APPLY
    (
        SELECT 
            TOP 1 
            Percentage
        FROM
            tblStage1 t2
        WHERE 
            t2.Project = dl.Project
            AND 
            t2.Date <= dl.MonthStart
        ORDER BY 
            t2.Date DESC
    ) pct

Question 3

A really easy way would be to create a temp table with the month values of each month you need then doing a left join on that table.

Out of curiosity - why is August 1 instead of 0?