Yes, a 32.768 kHz crystal is often referred to as a 32 kHz crystal. I think it's rather sloppy personally, as it only costs another 4 characters to make certain of avoiding any potential confusion.
Yes, they will all work. Unless you need the frequency to be absolutely spot on then e.g. 22pF will do.
If you look at the notes in the datasheet it says they are for design guidance only, for exact values consult the crystal manufacturer. So the crystal datasheet is the one to go from. As Endolith says the microcontroller doesn't affect the correct load capacitance value (well maybe a tiny bit with the pin capacitance)
I agree with Olin that the 16F84A is an antique - if you grab one of the newer PICs you will give yourself far more options. The PIC16F1828 is a nice part, internal RC/PLL up to 32MHz and loads of nice peripherals. Probably won't be much more than the 16F84A, may even be cheaper.
There is much misinformation out there on this. The load capacitance specified for the crystal is what should be across the crystal from the crystal's point of view. As a first reaction, that is the series combination of the two caps C2 and C4.
However, there is a lot more going on there and various (usually invalid) assumption are embodied in this common knee-jerk formula. Instead of rules of thumb, look at what is really going on. Two things that can significantly change the apparent capacitive load across the crystal are stray capacitance and the impedance of the driver driving the input signal to the crystal.
Consider how this circuit looks if the impedance of the crystal driver output is 0 (perfect voltage source). The cap on that output is irrelevant, and the load seen by the crystal is the full output cap. To get some idea what impedance range this is relevant for, consider that the impedance magnitude of 22 pF at 16 MHz is 450 Ω. Is the impedance of the output that is driving the crystal 450 Ω or less? Quite possibly yes. However, this is rarely specified, so you have to guess.
You also have to guess what the stray capacitance might be. I'd say you're 5 pF figure is reasonable, given a reasonable layout. So if the crystal driver has infinite impedance, the capacitive load on the crystal is 16 pF. If it has 0 impedance, the load on the crystal is 27 pF. The reality is of course somewhere in between.
In reality, for normal typical microcontroller crystals somewhere around 10 MHz (factor of 2 is close enough), 22 pF caps on both sides is a good bet.
The point to all this is that there are a lot of unknowns you can't account for. This is one place you can't analyze down to small detail. If two 22 pF caps seems to get you in the ballpark, just go with that. That's what I do, and I haven't had any problems. Fortunately crystals have a very sharp frequency curve and a factor of 2 either way from the specified load capacitance is usually fine. If you need better than a few 10s of PPM, then you probably need to do some tests with carefully calibrated instruments.
Best Answer
Hobbyists don't always get the details correct, if it works they are happy. Follow the vendor recommendations.
The pull-ups are not only not needed, they can potentially cause harm since the DS1302 has internal pull-downs. I would guess that the author was accustomed to seeing pull-ups for I2C interfaces and thought that this was similar. The DS1302 doesn't use I2C or SPI, it is a custom interface.