Atmosphere's can escape by several different mechanisms, so there's no single formula which you can apply everywhere. But a good first approach is to just look at the loss due to the fastest particles escaping (called thermal or 'Jeans escape').

In a planetary atmosphere, there is a certain height (called the exobase, or base of the exosphere), below which molecules collide frequently enough to establish thermal equilibrium, and above which the mean free path is greater than the scale height (so a molecule thrown upwards with enough speed has a good chance of escaping without hitting another one). Molecules in a gas at a given temperature have a Maxwell distribution of speeds. Mathematically, this distribution extends to infinite velocities, but the dropoff happens so fast that in practice there are very few molecules with velocities greater than about four times the mean value, where the mean value is given by


where R is the gas constant, T is temperature in Kelvins, and M is the molar mass of the molecule.

If the mean value is more than 1/4 of the escape speed:


where G is the gravitational constant, M is the mass of the planet (yeah, sorry, same capital M referring to different things -- use whatever other notation you want), and r is its radius,

then this means an appreciable number of molecules will be moving faster than the escape speed, and the planet will lose its atmosphere very quickly. Beyond that, there is an exponential increase in the atmosphere's lifetime. If the ratio of escape speed to mean speed is 4, then it may last thousands of years. 5, millions of years, and 6, billions of years.

This is a rough rule of thumb that works fairly well for most purposes. I made a spreadsheet to calculate this easily in another thread where this discussion popped up -- see here. But if you want to be more precise, especially in the sense of actually calculating the mass loss rate, or flux of molecules evaporating from the atmosphere, then you can use the Jeans formula for the escape rate (atoms per square meter per second). This is



Where N_ex is the number density at the exobase, and λ_esc is the escape parameter,
(v_e/v_o)^2

If you apply this to atomic hydrogen in Earth's atmosphere, you'll get a flux of something like 6x10¹¹ atoms per square meter per second, which is comparable to the rate at which this gas is diffused to the upper atmosphere. I.e. it escapes the exosphere at almost the same rate as it gets there in the first place.

The greenhouse effect doesn't work that way. If you have some small transparent container of CO2, it won't be of any different temperature than the same container filled with N2.

Instead, the greenhouse effect works by absorbing some of the planet's outgoing thermal flux, and redirecting some of that back downwards. This causes a change in the vertical temperature profile -- heating the surface and lower atmosphere, and cooling the upper atmosphere (because less thermal flux gets there, and with more greenhouse gas these upper layers also radiate thermally more efficiently).

The physics of this is conceptually simple, but is very complicated in practice. It requires computational methods, integrating the absorption and emission of flux over the whole vertical structure of the atmosphere. A good treatment of this can be found in an atmospheric science text, but I won't go in into such depth here. In a more simple model, where the atmosphere is split up into layers, with each layer defined to have one extinction thickness (it absorbs 100% of the outgoing flux), we obtain the following formula for the ground temperature:



Where τ is the number of extinction thicknesses, and T_E is the equilibrium temperature calculated by the flux balance for a blackbody,


where L* is the star's luminosity, AB is the albedo of the planet, a is the planet's semi-major axis distance, and sigma is the Stefan-Boltzmann constant.

Finally, the temperature at the i'th layer is just i^1/4T_E.

You're right that the greenhouse effect will depend on the type of star, but this basically just shifts the incoming spectrum. Most stellar spectra approximate a blackbody spectrum, and for most cases that we might be interested in, this spectrum doesn't overlap much with the planet's emission. So if we're interested in a planet whose atmospheric gases are essentially transparent to that spectrum, then the star's temperature will just be reflected in L*.

This is another really complicated process. I hesitate to use a general formula for it because there can be so many factors that influence it (e.g. with the Earth-Moon system, it varies due to changes in the dissipative efficiency within the oceans). But in general, the locking time scales with the sixth power of orbital distance, so it's a very strong dependence. There's a formula on wikipedia if you want to use that.

Other formulas you might use are as follows (from Planetary Sciences, by Pater and Lissauer).



The rate of change of angular momentum (or torque), is



Where subscripts 1 and 2 represent the primary body and its satellite, respectively. k_T is the tidal Love number, Q is the dissipation factor, R is the body's radius, r is the orbital distance, omega is the angular rotation rate and n is the orbital angular velocity. (So the sign of omega minus n represents whether the satellite orbits faster or slower than the primary rotates, and the sign(x) function turns this into a +1 or -1 accordingly.)

This torque (if none other are present) will make the orbit expand or contract. The rate at which this happens is



If you specifically want energy, this is transferred by the torque according to n (the orbital angular velocity) times the angular momentum. Or, dE/dL = n.



It generally doesn't, at least not by tidal forces. Tidal force will only cause orbits to collapse if sign(omega - n) = -1. So, if the satellite orbits ahead of the nearest bulge it raises on the primary, or if the satellite orbits retrograde. And for interplanetary distances, tidal forces is trivial, since it follows the inverse cube power of distance.

For looking at long term system evolution and instability, other effects are more important. For planets, the most important things are perturbations by the other planets. There was an awesome lecture on that topic posted here a while back if you're interested. And for small bodies, there are other effects like radiation pressure, PR drag, Yarkovsky effect, YORP, etc.