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I am interested to know if any particular slice of sky contains more galaxies or more stars (in the milky way, although would be interested to factor in extra-galactic stars, and even rogue planets)
How does this vary across the direction of the sky relative to the core of the milky way?
It depends which direction you look in and down to what brightness limit and in what waveband.
Roughly speaking there are about $10^{11}$ stars in the Milky Way and a similar number of galaxies in the observable universe. However, whilst the galaxies are roughly uniformly distributed, the stars are very much concentrated towards the galactic plane and centre. There is also a concentration of obscuring dust towards the galactic plane that prevents the observation of distant objects. Finally, whilst more and more galaxies are revealed in deeper observations, the number of faint stars eventually flattens off, because the Milky Way stars are limited in how far away they can be.
The net result is that: shallow observations (e.g. your naked eye) are dominated by stars in all directions; deep observations towards the galactic plane are also dominated by stars; but deep observations out of the galactic plane are dominated by galaxies. There are also many more red stars than blue stars, but also many more red galaxies than blue galaxies. Redder wavebands are also less affected by dust extinction so this would play a lesser role in any comparison.
In terms of some basic numbers, here is a galaxy number counts plot at high galactic latitudes in both the B- and K-bands (near infrared), taken from here. The units on the y-axis are the log of number per square degree per 0.5 magnitude interval for the B-Band data and per unit magnitude interval for the K-band data. The cross marks galaxy counts in the Hubble Ultra DeepField.
Still trying to find some good plots of stellar numbers, preferably as a function of galactic latitude, but at high galactic latitudes, the slope of the relationship is shallower and it crosses the galaxy curve at around $B=20$. i.e. Fainter than this, galaxies dominate and by orders of magnitude by the time you get to the limits of the Hubble Ultradeep field.
