I First question
The meaning of "entrainer" in this sentence is neither "encourage" nor "coach".
From the TLFi (definitions have been changed to bold type)
I.− [L'idée dominante est celle de mouvement imprimé à qqc. ou à qqn]
C.− Au fig.
c) [Le suj. désigne une chose abstr.] Emporter, pousser quelqu'un vers quelqu'un ou vers quelque chose sous l'effet d'une influence irrésistible
2. [Le suj. désigne une pers.] Amener quelqu'un à agir, à faire quelque chose en exerçant sur lui une contrainte, une pression morale
► Comment le supposez-vous en même temps assez amoureux de la guerre pour y entraîner ses chefs, qui ne la veulent pas? (Renan, Drames philos.,Prêtre Némi, 1885, III, 1, p. 569).
► Il sait comment on entraîne les peuples et comment on les fanatise (Coppée, Bonne souffr.,1898, p. 105)
► Il n'y a qu'un mot à dire à ces impuissants de la liquidation : « ce qui est fait est fait », et les entraîner de force à de nouveaux destins. Mounier, Traité du caractère,1946, p. 453.
The essential of the definition, "Amener quelqu'un à agir, à faire quelque chose en exerçant sur lui une contrainte, une pression morale", means "to bring someone to act or to do something, this being achieved by exerting on them a constraint or moral pressure". Moral pressure is not relevant here. The constraint might have consisted in encouragements (not much of a constraint but still not negligible), however it is not likely that you can impart the courage to laugh, that is exhort someone to laugh in explaining that there is nothing to be afraid of. The constraint was possibly of a different nature: it could have consisted in the stimulation of their ability to emulate, which I think more probable. Nonetheless, this is not told and all that is said is that she brought him to laugh. In French this would be synonymously put as (from the definition) "…combien de fois elle l'avait amené à rire…".
ref. Memoirs of Louis XIV and His Court and of the Regency ... I had brought him to laugh at his own weakness, and even to take jokes without caring for them.
The sense "II. B" in the TLFi, which would be "to coach", does not seem to apply.
B.− P. ext. [Le suj. désigne une pers.] Former, soumettre quelqu'un à une activité physique ou intellectuelle de manière à créer une aptitude ou une habitude.
It implies a rather methodic activity ("former", "formation") which is not likely here.
II Second question
As seen in the first part, "entrainer" has a wider meaning and a perusal of the entry in the TLFi should make you aware of its scope. The particular sense I'd retain in this sentence ("Édouard avait quand…") has been mentioned in "I". The second part of this second question can't be answered in a scholarly way before a lengthy comparative study of the two languages; an approximate answer from the mouth of a professional translator could possibly have some value. Personally, I can't confirm or infirm this contention, I do not have enough experience. What I can assert is that expression, in any language I presume, but certainly in French and in English, does not consist in formulating in all circumstances the most detailed statement applying to a situation. For instance, in English, you can talk of such and such a number of fighters put out of action, for instance, without saying precisely what is the number of the dead and the wounded; nothing keeps you then to become more precise and produce those numbers; you can as well start with those numbers and forget about the general statement. This is similar to this question of "entrainer quelqu'un à rire".