Photosystem 1 and 2; P680/P700; Chlorophyll a/b
Regarding your questions #1 ("Is the primary pigment reaction centre in both photosystems a pair of chlorophyll a molecules?") and #3 ("How can it be that it absorbs at a different wavelength in the two photosystems if it is the same molecule?"):

Both reaction centers in Photosystem I and Photosystem II contain only chlorophyll a. According to Lodish (Molecular Cell Biology):


  As in the bacterial reaction center, at the center of each chloroplast photosystem is a pair of specialized reactioncenter chlorophyll a molecules, which are capable of undergoing light-driven electron transfer. The chlorophylls in the two reaction centers differ in their light-absorption maxima because of differences in their protein environment. For this reason, the reaction-center chlorophylls are often denoted P680 (PSII) and P700 (PSI). (emphases mine)


Regarding your question #2 ("If so, what is chlorophyll b? Is it simply an accessory pigment?"):

Chlorophyll b has a different molecular structure (and, therefore, a different absorption spectrum) and it's less abundant in the chloroplast than chlorophyll a. Here is an image depicting their molecular structures and absorption spectra:


(a) Chlorophyll a, (b) chlorophyll b, and (c) β-carotene are hydrophobic organic pigments found in the thylakoid membrane. Chlorophyll a and b, which are identical except for the part indicated in the red box, are responsible for the green color of leaves. β-carotene is responsible for the orange color in carrots. Each pigment has (d) a unique absorbance spectrum. Source: "OpenStax College, The Light-Dependent Reactions of Photosynthesis. October 16, 2013." 

Both chlorophyll a and b are present in the antenna complex. Still according to Lodish:


  As in photosynthetic bacteria, each reaction center has an associated antenna that consists of a group of light-harvesting complexes (LHCs); the LHCs associated with PSII and PSI contain different proteins.


However, most of chlorophyll b is associated to PSII antenna complex, not PSI, which is one of the main differences between PSII and PSI.

Source: Lodish, H. (2000). Molecular cell biology. 4th ed. New York: W.H. Freeman.