All three of the problems you enumerate are real, and, probably, universal - even (and perhaps especially!) amongst professionals.

When dealing with late entries, I simply stop and rehearse the quick renewal of breath followed by a note or two right on beat. It takes time for some to learn how to breath without disrupting beat or rhythm; or, more properly, how to breath in rhythm. This is not unlike training piano or organ students to learn their notes so well that they don't disrupt rhythm whilst finding (or correcting!) wrong ones (it being well understood that a wrong rhythm moment is an evil equal to a wrong note moment). The fundamental problem here is identity with a metrical and rhythmic continuum. This has to be rehearsed and drilled in.

Akin to the above is the fear of the role of silence in musical performance, which often leads to quick, gasping, and nervous breathing - as if breathing were not an inherent element within the flow of music, as if the music stops and we take a breath and then the music continues. No! Rightly portioned breathing is of equal importance as rightly portioned sound. I teach my singers that one plays or sings a silence just as surely as he or she plays or sings a note. The silence and the note both must be performed with perfect rhythmic and metrical continuity. Practice stopping your singers at random spots, enjoying five seconds of silence, and proceeding with performing the sound as though nothing had happened, as though the silence had been a deliberate and inherent (metrical!) element of the performance. (And! DO NOT allow them to breath during this hiatus - hold their breath, their diaphragms and voices in perfect stasis and renew the sound with perfect intonation! WITHOUT GULPING. Without tension in head or neck muscles. Without bodily movement or reflex. Tone production involves basically 1) diaphragm, 2) vocal chords and apparatus, 3) shaping of the mouth cavity for desired vowel quality. The rest of the body should be in an erect posture, but without tension or stiffness - especially of the neck and head.

Also, there are many places at which a break occurs where a replenishing of breath is really unnecessary. Here singers can be rehearsed on gracefully ending a word or note and, with zero breathing and zero disruption of the vocal apparatus and pitch preparedness, after a very brief musical pause (cued by the choirmaster) renewing the sound. In these instances people can be taught that every time there is a break they really do not have to destroy their vocal focus in the act of gulping for air. Proper breath control and diaphragm-supported wind pressure are the key to success in this area.

On your last problem: faulty intonation following a breath is a common problem. This is related to what I said above about not jarring vocal or diaphragm posture just to take a breath. Keep those 'tummies' in. As the breath is expelled draw those tummies in as the diaphragm pressures out the air. Breathe anew without involving a lot of any pooching of the tummy in and out. This disruption of good vocal physique and posture is responsible for much of the sloppy phonation that follows a breath.

I'm sure that some of our vocal PhDs, or professional singers could state this better than I, or even correct me. But these are my off-the-cuff observations based on my experience.

OH! And I forgot another very important matter! How do your people stand, and, more importantly, sit, while they are singing!? When singing one should insist with great insistence that his or her singers sit up straight and not slouch against the back of the pew or chair. Never allow people to sing leaning against the back of the chair. I tell my singers to 'sit stand' - this means (when they are sitting) that from their 'bottom' up they are maintaining a 'standing' posture. Accurate pitch, accurate blend, accurate intonation, etc., are impossible if people's posture does not permit it. Slouching (as is so so common, and so so permitted or overlooked) should be absolutely forbidden in rehearsal and performance. This is something that has to be constantly harped on and expected.

How many singers are in your schola, Carl?

Some quick reminders for singers-
Breathe only for the length of the phrase
Take your voice out sooner than later when "sneaking" a breath

In Gregorian chant, if you have two or more singers in the same octave, (ideally at least three, in my experiences) the techiniquee of taking one's voice out on a note or two works very well, and often better than breathing between notes or out of a note of length. That helps with pitch and late or uncoordinated entrances.

I don't feel it ever necessary to augment advice from MACW. I do offer the simple reminder that the concept of a "catch breath" needs explication. The greatest reminder is that an open trachea for a fraction of a second enables a singer (or flutist, who loses half of exhalations every moment) to replenish without concerted effort a supply necessary for appropriate phrasing. A singer's catch breath is not a swimmer's catch breath.