Measuring productivity with the Törnqvist index
The formula you have above to solve for multifactor productivity (or total factor productivity) is correct; however, after reading your post, you are interpreting it differently.   

The terms with the natural logs in the formula is asking you to obtain the rate of change (growth rates) [same as the difference in logs seen in formula] of a particular period for capital, labor , and output and the average cost shares for labor and capital.     The term on the left indicates the log difference of MFP which is the same as the rate of change (growth rate) of MFP.

To answer your first question:
 If you are calculating firm level productivity you should be using firm level labor input (number of hours worked of all persons), capital input , output data.     

For your second question:
The number of hours should not be used as labor input cost share. That term would be labor compensation or total wages.   

The term ~s is actually the two period average cost share of labor input [cost share of labor input is the labor cost (or labor compensation) divided by the cost of both labor and capital (which should be the same as total revenue/output/sales)]. 

-**So your statement is correct; you should multiply the hours worked with an average base wage to get the labor cost in monetary units. total employment costs should also be the same and work. **-

For average capital cost share, the formula provided is actually showing you a shortcut in coming up with average capital cost share.   Just take the average labor cost share and subtract it by 1.  

For your third question:
Capital should reflect the real value all physical assets related to the firm’s final output which should include the use of:

Equipment

Structures

Inventory- items that are in the process of being prepared for sale

Intellectual property products (software, R&amp;D, artistic originals)

Land 

These fixed production assets costs should include depreciation. 

To calculate MFP growth in 1987-1988 (or 1988%)



Real output % = 4.3 

Labor input% = 3.2

Capital input%= 3.9

Average labor cost share =  ( .663 + .671) /2  = .667

Average capital cost share= (1- .667) = .333 [shortcut]

4.3 – (.667*3.2) – (.333*3.9) = .87

Actual value of MFP% is  =  .8 

*note that doing it this way in few periods will cause rounding issues that may show a .1 (tenth decimal) difference.  But in most periods you will be right on target.