Is it possible to execute a stored procedure over a set without using a cursor?

If you're only using SQL Server 2005 or newer, don't care about backwards compatibility and can convert your code to be in a User-Defined Function (rather than a stored proc) then you can use the new "CROSS APPLY" operator, which does use a syntax very similar to what you want. I found here a short intro (of course, you can also read the BOLs and MSDN)

Supposing your SP returns a single value named out_int, your example could be rewritten as:

SELECT T.id, UDF.out_int
FROM 
    Table1 T
CROSS APPLY
    dbo.fn_My_UDF(T.id) AS UDF

This will retrieve each "id" from Table1 and use it to call fn_My_UDF, the result of which will appear in the final result-set besides the original parameter.

A variat of "CROSS APPLY" is "OUTER APPLY". They are equivalents of "INNER JOIN" and "LEFT JOIN", but work on joining a table and a UDF (and calling the second at the same time).

If you must (by explicit order of the pointy-haired boss) use SPs insead, well - bad luck! You'll have to keep with cursors, or try cheating a bit: change the code into UDFs, and create wrapper SPs :D.

Yes. If you have a column you can use within the table to mark the ones processed, you can use WHILE EXISTS:

DECLARE @Id int
WHILE EXISTS(SELECT * FROM Table1 WHERE Processed='N')
BEGIN
 SELECT Top 1 @Id = id from Table1 WHERE Procesed='N'
 EXEC dbo.Sproc @Id
 UPDATE Table1 SET Processed = 'Y' WHERE Id = @Id
END

Alternately, dump the ids into a temp table or table variable and delete when finished:

DECLARE @HoldTable table (Id int PRIMARY KEY)
DECLARE @Id int
INSERT INTO @HoldTable SELECT Id FROM Table1
WHILE EXISTS(SELECT * FROM @HoldTable)
BEGIN
 SELECT @Id = id from @HoldTable
 EXEC dbo.Sproc @Id
 DELETE FROM @HoldTable where Id = @Id
END