Why do insects have very long pre reproductive periods and very short reproductive periods? Is there a common characteristic that can indicate the duration of these stages or offer an explanation for the same?
I actually see two different questions in your question. The first is "what determines the duration of the larval stage ?", the second is "why is this duration longer that the adult stage ?"
A) What determines the duration of developmental stages?
First, you must note that not all insects follow the same development process. Holometabolous insects (e.g. butterflies) have four life stages: egg, larva (with several instars), pupa and adult (or imago). On the other hand, hemimetabolous (e.g. bugs) only have three life stages, and some insects are ametabolous, i.e. no metamorphosis. Development time thus greatly depends on the mode of development. Of course, there can also be a huge variation within insects that have the same mode of development.
What else determines the duration of developmental stages ? (i.e. eggs/larva/pupa) A large combination of factors. Before they reach the adult stage, insects need to accumulate enough energy to undergo every developmental step and become mature. Resource quality and availability of course influence developmental "speed", but abiotic parameters such as temperature or humidity are also of prime importance. Actually, development time of many insects can be measured in degree days, which corresponds to the accumulation of temperature over time. For instance, larval-pupal development of the codling moth Cydia pomonella can go from 1 to about 6 months, according to temperature conditions.
B) Do insects have longer pre-reproductive periods that reproductive periods ?
- It depends. There is a huge variation a) in the lifespan of all insect species, b) in the relative length of their developmental stages.
Some insects indeed have a very short adult life stage, relatively to their developmental stages. Mayflies are a classical example, but it's also true for many butterflies and moths of for mosquitoes, for instance.
On the other hand, it's not true for many other insects. Many Coleoptera (e.g. beetles, cockroaches) can have several-month long adult lives, sometimes way longer that their life as a larva. Bees are a more extreme example: bee queens need three weeks to become fertile, and can then reproduce during several years (same for ants).
C) Why do s̲o̲m̲e̲ insects have short reproductive periods ?
To my knowledge, the reasons are mainly ecological. Insects that have short adult lives are often fragile, flying insects that disperse a lot. They face an increased risk of predation as well as environmental fluctuations (weather, etc.) against which they are poorly protected. Besides their resources (e.g. sugar) may be scarce or poor-quality - some even do not have the ability to feed, e.g. mayflies.
What about evolutionary reasons? Developmental stages need some time because, as I said earlier, insects need to acquire a lot of energy to pass all developmental steps. I didn't even talk about metamorphosis, which is a complete cellular re-organization -that takes some time. On the other hand, adults just need to mate and/or lay all their eggs before they die. They don't need to live too long as they'll probably die soon anyway. Think about mayflies: they are less protected as free flying adults that as underwater-protected nymphae. And they'd better try to maximize egg laying in a reduced time.
Besides, note that there is a difference between potential lifespan and realized lifespan. For instance, parasitoid wasps can live up to 30 days in safe, food-abundant conditions - in the wild however, they often live just a few days.
Harvey, J. A., Cloutier, J., Visser, B., Ellers, J., Wäckers, F. L., & Gols, R. (2012). The effect of different dietary sugars and honey on longevity and fecundity in two hyperparasitoid wasps. Journal of insect physiology, 58(6), 816-823.
Müller, T., & Müller, C. (2015). Behavioural phenotypes over the lifetime of a holometabolous insect. Frontiers in Zoology, 12(1), S8.
Rafoss T, Saethre M-G (2003) Spatial and temporal distribution of bioclimatic potential for the codling moth and the Colorado potato beetle in Norway: model predictions versus climate and field data from the 1990s. Agric Forest Entomol 5:75–85