How are the dominance and additive effects reflected in the mean and variance of the offsprings of two individuals with known phenotype?
What is the expected phenotypic trait of an offspring given the phenotypic trait of its parents?


The expected trait of the offspring is equal to the mean of the parents traits.

For this I assumed we don't have more information about the genetic of the trait (typically, without knowing the directionality of dominance and number).


  What is the expected phenotypic trait of the offspring population given the phenotypic trait of the parent population?


As a consequence of the above, the expected mean phenotypic trait of the offspring population is equal to the mean of the parent population (in absence of selection).


  What is the expected variance in the offspring population?


No drift, no selection

In absence of drift and selection (and environmental changes), the variance of the offspring population equals the variance of the parent population.

Directional selection only

In presence of directional selection, please have a look at How does Natural Selection shape Genetic Variation? and ask follow-up question if needed.

Drift only

In presence of drift only, the heterozygosity is reduced by a factor $\frac{1}{2N}$, where $N$ is the population size. The loss of genetic variance is then dependent on (1) the frequency, (2) the dominance and (3) the effect size of alleles at loci causing this genetic variance. There is therefore no simple answer. The only generalization that can be done is that (1) the expected variance of the offspring population is lower than the variance of the parent population and (2) the smaller the population size, the higher is the expected loss of genetic variance 

General Note

My feeling about quantitative genetics (but I am not really an expert in that field) is that it is often frustrating to see that generalization is often impossible and general solution to recursive equations often don't exist implying that predictions to more than one generation must be calculated through iteration.

I suspect that an expert in quantitative genetics would be able to say much more than me in this post.

The book Introduction to Quantitative Genetics (by Falconer and Mackay) is probably still the best book to increase your knowledge in quantitative genetics.