Is my reasoning about economic growth and environmental sustainability correct?
It is on a correct track but I would not say it is 100% correct as you miss the main source of growth which is technological progress. Consider the following Cobb-Douglas production function with all 4 main factors of production, capital ($K$), human capital ($H$), labor ($L$), land ($M$) then the standard production function would look something like:

$$F(K,H,L,M) = A K^{\alpha}H^{\beta}L^{\gamma}M^{1 - \alpha - \beta - \gamma}$$ 

Well you are correct to say that production $F$ increases if either more capital is put to work, or more labor hours are employed ($K$ or $L$ increases). However, there can be limits to how far you can push capital accumulation and still be sustainable.  Machines after all need some raw materials. Also, people only have 24 hours a day so there is a limit there. Land is for all practical purposes fixed (at least before we can make an effective use of real estate on moon or mars - and I am saying that just half-jokingly). However, you missed one important factor of production: human capital. That is things like better education which as far as I can see can increase almost without bounds.

Second thing you missed is $A$ that is the total factor productivity, which makes any factor more productive and loosely speaking it is the technology avaiable. For example, thanks to our better understanding the computer technology modern computers are more faster than the old ones yet they are produced with the same amount resources (or sometimes even less resources - think of the old bus size punch card computers). This does not apply just to capital, thanks to telecomunication technology you can make better use of your human capital, thanks to better management techniques better use of labor, thanks to agricultural techniques like crop rotation better use of the same land. 

Also some things that were not correct in your answer:

Comparative advantage (unless you are considering some dynamic endogenous growth effects) generally does not lead to higher economic growth. It will increase the efficiency so GDP will have one time jump from lets say 100 to 200, and sometimes the jump may take several years so on yearly accounts it looks like growth, but its really just increase in level - that is still good but its not really long run growth.