BQ24450 or UC3906 Max Allowed voltage

Yes, either IC can charge a 24V battery.  The maximum rating on the IN pin is 40V.  OUT must be lower than IN for the linear charger to work properly.

Either the UC3906 or BQ24450 are available and orderable.  For new designs, I would recommend the bq24450.

these days i'm working on BQ24650 . It's adjustable . I prefer you to use it.

First, let me complete the circuit in figure 9 which has been truncated on the left of the screen: add a diode from Charger’s Primary Power Source to D1’s cathode and connect that point (D1 cathode/Supply Input) to pin 5 of the IC.  Add a capacitor from that point to ground labeled C1.

R5-R8 are sized in the usual way to provide the proper pre-charge, boost charge, full-charge. and battery cutoff points.

R3 and R4 are sized to provide enough base drive to Q1 when powered from either the pre-chg pin or the primary power source.

R1 and R2 and sized to limit the current through Q1 and the relay.

I cannot determine if a given transistor is ok for a given application.  You can calculate the power being dissipated in it in your system under your conditions and then calculate the amount of temperature rise based on how you will mount it and the airflow and ambient temperature etc.

C2 is sized as explained in the app note.  It is there to supply the load for as long as it takes for Q1 to turn off and the relay to close, as the input power is removed.

This circuit will disconnect the battery from the load if the battery voltage gets too low.  This simply sets a voltage level at which the cell will not be allowed to discharge past.

Vref is specified in the datasheet and is the internal, temperature varying reference voltage of 2.3V at 25C.

Yes, there are decimal points missing in the formulas but you were keen enough to catch that.  Yes, 95 should be 0.95, 9 should be 0.9, 25V should be 0.25V, and 025 should be .025.  The 'V' is the units of Volts to make the units work out in the formulas.

Note that these formulas in the app note directly correspond to those in the bq24450 datasheet.

V31 is the pack voltage at which charge will be started again after having previously finished (gone through all 3 phases). It is simply a voltage at pin 13 equal to 90% of the 2.3V reference. 

The 4 resistor around the IC (R5-R8) can be sized exactly the same way as in the bq24450 datasheet example design.  The VTH is the final discharge voltage that the relay will trip at.  Instead of trickle charging the pack, it will open the relay to disconnect the pack from the load.

I don't think you can use any of the equations in the app note, as none are applicable to the relay load disconnect circuit which uses 4 resistors not 2 or 3.  Instead, use the equations in the bq24450 datasheet design example.  Using those, I get: Rc = 46k, Rd = 662.5k, Rb = 16.6k, Ra = 484.6k.  But check my math.

Yes, you can use an NPN pass element.  The bq24450 datasheet provides a good explanation of different pass element topologies with some tradeoffs for each.

Yes, you have to use a pass element (transistor).  The UC3906 controls the charging by controlling this pass element.

Figure 9 is a simplified version of a basic UC3906 schematic.  They didn't draw all of the parts needed as that would make figure 9 too complicated.  So, they are simply building on other circuits introduced earlier in the app note and just showing what is new.