just saw this, and, though it is a bit old, I add my thoughts:
As Stelios already said correctly, you are referring to the climb angle, i.e. the pitch.
Do climb rates for a given flight mission stay constant in one phase of flight (e.g. until reaching 1500ft altitude) and then change at a specific flight level or are they continuously adjusted?
Normally they do.
As climb angle we choose around 15° at TO depending on TOW (12.5-20), 20° only when you are light and can climb very quick or when you're empty and want some extra fun, but beware of your PAX. So take 15° when in doubt, 10° is too low.
After 800ft AGL you will reduce pitch to 10°, reaching 220KIAS somewhere. On 3000ft AGL select CLB thrust with ATHR on and select speed mode of 250 (depending on your SOP) or activate AP, when using HDG/NAV mode. In HGD mode select speed 250-300 (dep. on TOW) and climb to cruise FL, accelerate to your CRZ speed (say, 0.78 MACH).
That's just a rule of thumb and depends on wind, weather conditions and FL.
Do given climb rates exist for (one arbitrary) given flight mission?
Is there a method to calculate the change of climb rates depending on altitude and weight?
That might be a complex function, so recommend using table from QRH.
The climb RATE (= the speed of climb) is a function of thrust, TOW (take-off weight) and pitch (and environmental parameters, Manu has attached a web site that gives more detailed insight into that) . You always will decide between max("best") climb RATE and max climb SPEED depending on if you
want to climb as high as possible in a given time, or climb as quick as possible to your target FL (flight level). Normaly max climb rate is the preferred choice.
Are there sources like papers, reports, manuals that will work for genuine citation?
You'll find this data spread in both QRH and FCOM.
(edited and removed unrelated infos, if you have further question, don't hesitate to ask)