One other thing that is important is that mass should be celebrated there at least twice a month regularly. And the hosts should be replenished regularly.

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Of course, that was during a time when there was no worldwide apostasy and every Catholic chapel had its own tabernacle with reliable sacraments.
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Certainly the "traveling" clause does not prohibit any layman from ever carrying consecrated hosts on his person, such as to deliver to someone in need, perhaps a priest.  It likely means that one should not make a habit of traveling around carrying hosts, especially when the trip has nothing to do with delivering the hosts.
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Even during the pre-Vatican II period it was recorded that there were situation in which laymen were deputed as extraordinary ministers of Holy Communion. 

What does "during the pre-Vatican II period" mean? I mean when does that period start?
And under what conditions was it sanctioned?
I believe there were no "extraordinary ministers of Holy Communion" pre-Vatican II.

What is your evidence for saying this?

Pre-Vatican II is any time before Vatican II. My evidence is from my parents catechism books which were based on the Baltimore Catechism and also anecdotal. The Chronicles of the Catholic Church in Lithuania makes reference to an incident in which some one (a lay person) was  allowed by the bishop to bring a consecrated host into a Soviet prison. There was another incident I heard of in which a woman was allowed to bring a consecrated host home with her to give to her elderly mother who was too ill to come to mass. Since the Soviet Union restricted access to information, the docuмents of Vatican II were not promulgated until the time of or close to the fall of the Soviet Union. 

Depends on what your definition of an extra ordinary minister is.

The way I understood it from childhood was that anyone who was delegated to take the Holy Eucharist  somewhere was an extra ordinary minister for that occasion.  It wasn't a designation for life.

This happened many times during the war.
Father Gereon Goldmann OFM was often an extra ordinary minister when he was in the German army and he later became a priest.  He true story is in his book The Shadow of His Wings. from Ignatius Press.

An emergency could be anytime a priest couldn't do it.

Speaking of keeping the Eucharist in your home, here is a reply to that question published in Zenit;

Q: Please advise whether extraordinary ministers of Holy Communion may keep the sacred species at home overnight in order to distribute to the housebound the next day. It would be inconvenient but certainly not very difficult for them to return to the church to collect their pyx the following day. — P.H., St. John’s, Antigua and Barbuda
A: The short answer would be no. Canon law is quite clear on this point. Thus:
“Canon 934 §1. The Most Holy Eucharist:
“1/ must be reserved in the cathedral church or its equivalent, in every parish church, and in a church or oratory connected to the house of a religious institute or society of apostolic life;
“2/ can be reserved in the chapel of the bishop and, with the permission of the local ordinary, in other churches, oratories, and chapels.
Ҥ2. In sacred places where the Most Holy Eucharist is reserved, there must always be someone responsible for it and, insofar as possible, a priest is to celebrate Mass there at least twice a month.
“Canon 935. No one is permitted to keep the Eucharist on one’s person or to carry it around, unless pastoral necessity urges it and the prescripts of the diocesan bishop are observed.”
With respect to the tabernacle the law states:
“Canon 938 §1. The Most Holy Eucharist is to be reserved habitually in only one tabernacle of a church or oratory.
Ҥ2. The tabernacle in which the Most Holy Eucharist is reserved is to be situated in some part of the church or oratory which is distinguished, conspicuous, beautifully decorated, and suitable for prayer.
Ҥ3. The tabernacle in which the Most Holy Eucharist is reserved habitually is to be immovable, made of solid and opaque material, and locked in such a way that the danger of profanation is avoided as much as possible.
Ҥ4. For a grave cause, it is permitted to reserve the Most Holy Eucharist in some other fitting and more secure place, especially at night.
“§5. The person responsible for the church or oratory is to take care that the key of the tabernacle in which the Most Holy Eucharist is reserved is safeguarded most diligently.”
This is further clarified in the 2004 instruction Redemptionis Sacramentum:
“[131.] Apart from the prescriptions of canon 934 § 1, it is forbidden to reserve the Blessed Sacrament in a place that is not subject in a secure way to the authority of the diocesan Bishop, or where there is a danger of profanation. Where such is the case, the diocesan Bishop should immediately revoke any permission for reservation of the Eucharist that may already have been granted.
“[132.] No one may carry the Most Holy Eucharist to his or her home, or to any other place contrary to the norm of law. It should also be borne in mind that removing or retaining the consecrated species for a sacrilegious purpose or casting them away are graviora delicta, the absolution of which is reserved to the Congregation for the Doctrine of the Faith.
“[133.] A Priest or Deacon, or an extraordinary minister who takes the Most Holy Eucharist when an ordained minister is absent or impeded in order to administer it as Communion for a sick person, should go insofar as possible directly from the place where the Sacrament is reserved to the sick person’s home, leaving aside any profane business so that any danger of profanation may be avoided and the greatest reverence for the Body of Christ may be ensured. Furthermore the Rite for the administration of Communion to the sick, as prescribed in the Roman Ritual, is always to be used.”
First of all, we see from Canon 934.2 that the Eucharist can be reserved in a bishop’s chapel. Therefore, not even a priest can conserve the Eucharist in his home without the bishop’s permission.
Although some bishops do grant this concession to priests, it is usually if the rectory has a room set aside as an oratory which fulfills the conditions mentioned in Canon 938.
However, Canon 935, while stating the general prohibition, does allow for the bishop to make some exceptions, even stable one, by means of issuing particular prescripts and norms.
There might be some situations in which an ordinary or extraordinary minister might need to keep the sacrament in the home or other place outside the church — for example, if the minister lives a great distance from the church and needs access to the Eucharist for the sick and dying. In such cases the bishop may grant the exception. However, the admonition of Redemptionis Sacramentum No. 133 to avoid all profane activities while carrying the Eucharist must be strictly observed.
In short, while allowing for special circuмstances, these exceptions are to be avoided.
There does not appear to be any special reason for an exception in the case presented by our reader, and therefore the usual practice should be followed.
Indeed, from the spiritual point of view, it would be better for the extraordinary ministers of Holy Communion to take the Eucharist from the hosts consecrated at the Mass they attend. In this way, although it is by no means a requirement, it could help those who are sick or housebound to experience a greater sense of communion with the worshipping community.

https://zenit.org/articles/reserving-the-eucharist-at-home/