How does PF or "pilot flying" and PNF or "pilot not flying" affect logging?
While the question is fairly broad, looking at 61.51 is a good place to start, as well as 61.57 for some of the currency stuff.
But let's try and work out some of your examples:
How should actual instrument be logged? Should a PNF be able to log actual time if they are monitoring during instrument conditions?
61.51(g) is what dictates the logging of instrument time:

(1) A person may log instrument time only for that flight time when the person operates the aircraft solely by reference to instruments under actual or simulated instrument flight conditions.
(2) An authorized instructor may log instrument time when conducting instrument flight instruction in actual instrument flight conditions.

So barring any legal interpretations out there or regulations I'm missing (note, I'm not an instrument rated pilot nor attorney), a strict reading of the reg would indicate that only an authorized instructor, conducting instrument flight instruction, can log instrument time when not operating the aircraft.
How should landings be logged? Should a PNF be able to log landings they were not at the controls for?
The short answer is no. Assuming the landings are being logged for currency reasons, 61.57(a)(1)(i) states that the person had to be the sole manipulator of the flight controls. 61.57(b)(1)(i) says the same thing, but for night currency.
How should approaches be logged?
Again, I'm guessing this would be for instrument currency, at which point 61.57(c)(1) would be in play.

Within the 6 calendar months preceding the month of the flight, that person performed and logged at least the following tasks and iterations in an airplane, powered-lift, helicopter, or airship, as appropriate, for the instrument rating privileges to be maintained in actual weather conditions, or under simulated conditions using a view-limiting device that involves having performed the following -
(i) Six instrument approaches.
(ii) Holding procedures and tasks.
(iii) Intercepting and tracking courses through the use of navigational electronic systems.

Again, I'm not an instrument pilot, but despite &quot;sole manipulator of the controls&quot; verbiage being absent in this particular paragraph, I'm guessing the FAA would construe it to be the same.
Of course, I'm more than happy to be corrected should I be wrong.