For example, if the rocket exhaust was lighter than the outside air, would this give more thrust(since the exhaust should not "sink" in the air)?
For example, if the rocket exhaust was lighter than the outside air, would this give more thrust (since the exhaust should not "sink" in the air)?
A short answer to this question is that the exhaust of a liquid rocket engine is actually less dense than the atmosphere, but the reason you state above is not one of the factors involved in launching a rocket and it makes no real contribution to engine thrust.
If you look at this Rocket Engines Wikipedia page, you will that there are a large numbers of parameters that determine how efficient a rocket engine is.
Space demands that we ignore the effects of nearly all of these parameters in favor of answering your particular question,
Using the space shuttles main engines as an example, rocket propellents require a high specific energy (energy per unit mass), because ideally all the reaction energy appears as kinetic energy of the exhaust gases. Exhaust velocity is the single most important performance parameter of an engine, on which vehicle performance depends.
Taking a typical rocket's propellent as liquid hydrogen and oxygen, and bearing in mind the heat at which it is ejected, the space shuttles main exhaust product was primarily water vapor as the hydrogen and oxygen combined.
Most of the exhaust in this picture of a Space Shuttle launch is steam.
Water vapor is less dense than dry air. At equivalent temperatures it is buoyant with respect to dry air, whereby the density of dry air at standard temperature and pressure is 1.27 g/L and water vapor at standard temperature and pressure has the much lower density of 0.804 g/L.
So, by choosing the propellents they did, on the basis of maximum performance, the question of the density of exhaust products compared to the density of the air was not a concern of the designers, as although exhaust density was low, due to the high speed at which it was ejected, the volume by density output of the engines resulted in a large amount of mass being ejected per second.
Liquid hydrogen (LH2) and oxygen (LOX, or LO2), are the most effective propellants in terms of exhaust velocity that have been widely used to date, though a few exotic combinations involving boron or liquid ozone are potentially somewhat better in theory if various practical problems could be solved.
As @CountTo10 said your question is not really a design consideration. But for curiosity's sake your question deserves an answer.
When, due to combustion, exhaust gases are created inside the exhaust chamber of the rocket, there are two forces that will act on the exhaust gas. One is the pressure difference between exhaust gases and ambient air created by combustion itself, which exists even if there is no gravity, and which tends to push the gas out from the rocket. Second is the hydrostatic pressure difference that arises due to difference in density of exhaust gas inside the rocket and ambient air outside, which requires action of gravity, and if exhaust gases are lighter then this tends to push ambient air into the rocket. No prize for guessing which one wins.