As stated in my answer to another question, the meaning of the term "canon lawyer" is not really clear. I will concentrate on the obligations for getting the correspondending academic degree (licenciate in canon law) and those for beeing an "advocate" in terms of the CIC. The advocates are those described by the first quote in the question.
licenciate in canon law
To get a licenciate in canon law you have to be enrolled at a ecclesiastical faculty. The essential requirements for this are defined in article 31 of the apostolic constitution Sapientia christiana (SC):
Art. 31. Facultates Ecclesiasticae omnibus patent, sive clericis sive laicis, qui, legitimo testimonio praediti, vitae moribus ac studiis praeviis idonei sint ut Facultati adscribantur.
Article 31. Ecclesiastical Faculties are open to all, whether ecclesiastics or laity, who can legally give testimony to leading a moral life and to having completed the previous studies appropriate to enrolling in the Faculty.
The phrase "ominibus [...], sive clericis sive laicis," ("all, whether ecclesiastics or laity,") clearly includes all Catholic but even all Christians (e.g. Protestants are laypeople). It is not clear, weather non-Christians are included or not. I guess they are included and "sive clericis sive laicis" only gives examples. In the new constitution Veritatis gaudium (VG), which was given on 8 December 2017 and will replace SC with the beginning of the 2018-2019 or 2019 academic year (article 88), the words ", sive clericis sive laicis," were obmitted in the new article 31 (the rest of the article stays the same).
The other questioned obligation is the moral one. The constitution says "qui, legitimo testimonio praediti, vitae moribus" ("who can legally give testimony to leading a moral life"). This testimony is defined by article 26 of the norms of application of VG (nearly identical for SC article 24):
Art. 26. § 1. Legitimum testimonium ad normam art. 31 Constitutionis:
1° de vitae moribus, pro clericis, sacrorum alumnis et personis consecratis datur a proprio Ordinario vel Hierarcha, Superiore vel ab eorum delegato; pro ceteris a quodam ecclesiastico;
Article 26. § 1. The legitimate documentation, mentioned in article 31 of the Constitution:
1. about the student’s moral life: for clergy, seminarians and Religious, this is to be given by their own Ordinary, Hierarch or Superior, or by one delegated by them; for all other persons, by some Church person;
So the student needs testimony about the his moral life "by some Church person". I do not know what exact requirements he have to meet.
The requirements for beeing the advocate of a party in a process is defined by can. 1484 CIC:
Can. 1483 The procurator and advocate must have attained the age of majority and be of good reputation; moreover, the advocate must be a Catholic unless the diocesan bishop permits otherwise, a doctor in canon law or otherwise truly expert, and approved by the same bishop.
So the only moral obligation is to "be of good reputation", what, I think, is less than a testimony about moral life. The advocate should be Catholic, but the bishop can dispense from that (e.g. if advocate is urgently needed). Additionally he has to be apporved by the bishop, but I did not found any oath he should swear (neither in the CIC nor in Dignitas Conubii), except of oaths of secrecy.