Do trees capture more CO2 during a fast growth stage?
Right, there are a lot of misconceptions about this. They are mostly to do with the difference between the magnitude of carbon storage and the rate of carbon uptake, and also the difference between whole site carbon storage and above ground biomass carbon storage. There is also ongoing research taking place which (as you might imagine) takes a very long time because of the life expectancy of most tree species.

The standard forester's response is: a hectare of new forest will tend to grow on a Sigmoid / logistic curve (well described by Paine et al. 2011) which looks like this:


This is based on an even aged stand (so, multiple trees, planted at the same time).

Interpreting this, we can see that the greatest rate of carbon uptake is between years 25 and 75 (for this site, species, etc.) but the greatest level of carbon storage is not really achieved until year 150+. You can read more about this in the infamous "Blue Book" by  Forest Research. 

Based on this alone, we would assume that mature forests do not sequester much more additional carbon per ha per year, but they do store a lot more than if we simply felled the forest to maximise carbon uptake. However... 

1) Below ground carbon (soil carbon, roots, stumps etc.) doesn't follow the same rules. Carbon can remain sequestered in soils (particularly peat) for very long periods and can continue to be laid down (depending on site conditions) more or less indefinitely. When soils are disturbed (for example by felling operations or drainage) they release carbon. So any forest disturbance will tend to release soil carbon.

2) It doesn't scale well to individual trees as pointed out in in Stephenson et al. (2014). This is because the total tree population per ha is of a variable size (tree mortality and thinning means that there are fewer trees per ha over time). 

This means we end up with a trade off between storing the carbon we have already sequestered and drawing down new carbon into the system. The degree to which the carbon sequestration rate decreases over time depends a lot on the forest soil, climate, species, and other local conditions. There is an academic discussion taking place over the difference between a simple measurement of above ground woody biomass (as shown above) and a whole-site carbon balance.

It can get quite complicated, and how you look at it can result in different results. Do you definitely want to model individual trees? Or are you thinking about forests in general?

Hope this helps.