First, consider that we exist in a Utopian society, no wars and no disease. A result of this Utopian society is that our technology has advanced considerably, improving healthcare and life expectancy (which averages around 100 years). Nobody is born with disabilities, there are zero cot-deaths, miscarriages or similar issues. Everyone is born healthy and is guaranteed to have a long, healthy life until they are roughly a century old.
Our advanced technology also gives us the means with which we can travel the immense distances of space and colonize other worlds. We have some form of FTL, so the travel time to these new worlds is measured in days or weeks. The worlds we colonize are effectively identical to Earth, and none of these worlds pose a threat to our survival (in terms of hostile native life or similar threats). This gives us all the land we need to support a population of 1 trillion. New worlds can only be colonized by a minimum of roughly 10,000 people. To keep administration simple, we can only colonize X worlds at one time, where X is the number of worlds that have at least 5 billion people living on them (so to start with we can only colonize 1 world, but once we have colonized that one, we can then colonize 2 worlds, and then after that 4 world and so on).
In this Utopian society, population growth rates are effectively governed by available space. The population on a world will no longer grow when it reaches 10 billion people (eg 2ish children per couple – whatever required to maintain the population but not exceed it). However a newly colonized world will frequently have birth rates within the 5-10 range – and this birth rate will slowly dwindle until the world has 10 billion people, at which point the population is fairly self sustaining.
Given these "perfect" conditions, and ignoring any possible interference from aliens or events that destroy entire worlds, how long would it take for 7 billion people to reach 1 trillion people.
Some additional details:
Consider this Utopian society to be very similar to a developed nation here on Earth for all concerns other than what I have described above; so similar gestation times, similar puberty ages etc.
Roughly a 50/50% male to female split for the population. People in this Utopian society wait until they are in their mid 20s before having kids, and consider a roughly 1-2% of people die from unnatural causes (inc suicides and accidents) and roughly 5% of people decide not to have any children.
Best Answer
Coming up with any sort of answer here will require a large number of assumptions. The most important factor is how the growth rate changes with population. The one solid value you've given is that the average lifespan is 100 years. I will make the simplifying assumption that this is from all causes, so includes the impact of unnatural deaths. That means that $1\%$ of the population dies every year. At $10^{10} = 10$ billion people, the growth rate is $0\%$, so the birth rate must be $1\%$ as well.
On the other end of the spectrum, you mention "5-10 range", which I assume means 5 to 10 children per couple for small colonies. 10 children amounts to a $5\%$ birthrate per year (5 kids per adult over a 100 year lifespan), and a growth rate of $5\% - 1\% = 4\%$. Let's assume that is the growth rate at $0$ population (note that it is an average birthrate, so the childless are accounted in that average).
To make the math simple, let's just assume a linear decrease in growthrate from $4\%$ at $0$ population to $0\%$ at $10$ billion population. So the yearly growth rate at population $P$ (measured in billons) is $$0.04 - \frac {0.04}{10}P$$ This gives a differential equation $$\frac {dP}{dt} = 0.04P - 0.004P^2$$
The solution to this equation is $$P = 10\frac {Re^{0.04t}}{1+Re^{0.04t}}$$ where $R = \frac {P_0}{10 - P_0}$ and $P_0$ is the initial population (in billions, at $t = 0$).
Now the less population a planet has, the faster it produces more. So planets should keep their own population as low as possible, to take advantage of the greater rate. Thus any planet with a population of $5$ billion should ship additional people away to its colony at a rate matching its growth ($2\%$, or $0.1$ billion a year), thus maintaining its population at the minimum necessary to have a colony. This adds an additional factor to the growthrate of the colony:
$$\frac {dP}{dt} = 0.1 + 0.04P - 0.004P^2$$
If we set $P = 0$ at $t = 0$ (since the parent colony is shipping $100$ million people a year, the requisite $10,000$ will be reached in the first day), the solution to this is $$P = 5 +5\sqrt{2}\frac{\sqrt 2 - 1 - (\sqrt 2 + 1)e^{-rt}}{\sqrt 2 - 1 + (\sqrt 2 + 1)e^{-rt}}$$ where $r = \frac{\sqrt 2}{25}$. Setting $P = 5$ and solving for $t$ gives that colony will need about $31$ years, $2$ months to reach $5$ billion in population.
At this point the total population has doubled, and now we have two planets at $5$ billion who can start two new colonies, repeating the process. I.e., the total population will double every $31$ years, $2$ months. To reach one trillion from $5$ billion requires $7.64$ doublings, so one should reach $1$ trillion by $238$ years (straight multiplying like that is not exactly accurate, but gives a quick estimation and is a relatively minor error compared to all the assumptions I've made).
Because Earth starts with $7$ billion instead of $5$, the first colony gets a head start, beginning at $2$ billion instead of $0$. That will several years off the initial $31$. But again, there is enough other error here to let that slide, so I will leave the estimate at $238$ years.