Is it true that animals that faced a severe bottleneck will have genetic deficiency for a long time? Does it mean they will all look the same for a long time?
A population bottleneck (see also founder effect), by definition is an event of sudden reduction of genetic diversity due to a sudden reduction in population size. Of course, such reduction of genetic diversity is not restricted to animals (unlike you seem to suggest) but to any population of living creatures.
Given that part of the phenotypic variance is caused by genetic variance, it means that after a bottleneck, phenotypic variance will be lower after the bottleneck than before the bottleneck indeed. However, a bottleneck might be coupled with a environmental change which could change the genetic variance in itself and the above might not be true anymore (see this post for an introduction to quantitative genetics).
Because, deleterious alleles tend to be recessive, a reduction in genetic diversity often causes a reduction in mean fitness. In humans, cases of founder effects have been shown to be associated with an increase in recessive genetic disease such as observed in Quebecois or in Amish.
The time needed to restore the genetic diversity very much depend upon the exact demography at play, as well as the mutation rate and selection coefficients. As a rule of thumb that many population geneticists use in numerical simulations; if $N$ is the population size at equilibrium (after bottleneck), then $10 \cdot N$ generations is enough to restore a good genetic diversity.