While your question asking about birds, reptiles and fish (oh my!) may be too broad, hopefully looking at frog oogenesis can show some differences in large offspring number v. small offspring number. Some frogs even give birth to live offspring (Iskandar et al. 2014).
Much of this explanation can be found in Developmental Biology, 6th edition by Gilbert SF.
In frogs, meiosis happens in stages (Developmental Biology):

This shows that different "cohorts" of oocytes form, develop, and are released, rather than a single group of mitotic events that are released gradually (as in humans). When the eggs have undergone vitellogenesis (yolk development), they are ready for fertilization.
As far as this question is concerned:
I am mainly interested in this to understand how the mother's chromosomes are represented in the offspring. Are all of her chromosomes guaranteed to be represented in at least one offspring?
If you're asking whether or not all of the mother's chromosomes will be represented in her total offspring, perhaps--with a large number of offspring (and a high ratio of fertilized eggs) the probability is very high.
No single offspring will contain all of its mother's chromosomes. Let's look at a diploid organism with two pairs of chromosomes: A and B. At the beginning of meiosis I, a cell will have one copy of both pairs:
A1 A2 B1 B2
Then comes synthesis, duplicating all chromosomes. Now we have two copies of each pair:
A1A1 A2A2 B1B1 B2B2
In prophase, chromosomes are condensed and the duplicated pairs connect in a process called synapsis:
A1A1-A2A2 B1B1-B2B2
In anaphase, the pairs are pulled apart so that each daughter cell has two copies of a single member of the pair:
daughter Y: A1A1 B2B2
daughter Z: A2A2 B1B1
Note that at this point it is not possible for either of the daughter cells to contain all of the mother's chromosomes.
With meiosis II, these copies are pulled apart, making four daughter cells:
daughter Y1: A1 B2
daughter Y2: A1 B2
daughter Z1: A2 B1
daughter Z2: A2 B1
These are now haploid, and await the male's sperm to make them diploid once more. Also note that daughters Y1/Y2 and Z1/Z2 are chromosomally identical (they can be different in practice with crossing over).
Here is a prettier representation of what I just described (Wikipedia):

The key is to remember that pairs of chromosomes and copies of chromosomes are different and to keep track of how they segregate during meiosis.
Edit: To address your clarification:
I am not talking about whether or not any of the daughter cells contains all of the mothers genetic material by itself, but rather whether every segment of genetic material is found in at least one offspring.
This is more of a probability question. If all oocytes mature, and are fertilized, the answer is clearly yes. However, not all oocytes are fertilized. Let's look at the same four daughter cells from my earlier example:
daughter Y1: A1 B2
daughter Y2: A1 B2
daughter Z1: A2 B1
daughter Z2: A2 B1
Let's say 2/4 (50%) are fertilized by the male frog. Going through all possible fertilizations, let's see how many combinations result in all of the mother's genetic material being passed down:
Y1, Y2 - No
Y1, Z1 - Yes
Y1, Z2 - Yes
Y2, Z1 - Yes
Y2, Z2 - Yes
Z1, Z2 - No
The answer to your question really depends on what proportion of oocytes make it to maturity, and what proportion of mature oocytes are fertilized. The probability of all the mother's genetic material being passed down increases with increasing fertilization ratio/egg number, so given the high number of offspring produced by many species of frog, I'd say that there is a good chance of all the mother's genetic material making it to the next generation, although it is not guaranteed.