What astronomical observations would give conclusive proof of alien life? There are several ways we are coming closer to answering the question "is there life elsewhere in the universe?". One is by first understanding very well the origin life on our own planet. Another is by trying to look for radio signals from extraterrestrial civilizations. Another is to try to search for habitable exoplanets, and eventually examine the habitable ones in detail via direct imaging. Another is too look for evidence of life or life that existed in the past right in our own solar system (for exmaple on Mars, Titan, Europa, etc). 

1) Have I missed any other astronomy-related approaches to answering this question?

2) Short of radio contact from an ET civilization, or a visit by aliens, or the discovery of life/past-life in our solar system, what would give conclusive evidence of alien life in the galaxy? 

For example:

In 20 or so years when we develop the technology to directly image exoplanets, we will be able to determine their chemical composition and characterize their structure. If astronmers who study exoplanets found what they are looking for -evidence of oxygen and other biomarker gases - this might be highly suggestive of life, but not conclusive evidence. What kinds of observations of an exoplanets could enable us to say with ceartainty "there is life on this planet"? Short of telescopic observations of truly unbelievable detail, I cannot think of a way that we can look at a planet and say "yes there is life on it, or no there is not" other than perhaps if seeing lights on the dark side of the planet. 

Edit: Actually, I forgot the obvious scenario where there are trees/grass on the exoplanet.  But I don't think the plant life is required to be a different color from the rest of the terrain, and a planet with aliens does not have to have plant life. So we might not be able to detect life visually.
Lots of frontier science lectures to watch on nearby topics here:
https://astrobiology.nasa.gov/seminars/

Your very question seems to be the hottest topic in astronomy today.

My impression is that if oxygen gas and methane were on the same exoplanet, then it certainly must be living biological processing creating that there. Nothing else could explain the coexistence of the two. And even if one of those two gasses could occur abiologically, and a range of exoplanets with one of them would be found, statistically one could deduce a frequency of planets which must have life, because the abiotic processes could only explain a certain maximum part of them.

It is a pitty that most exoplanets are in size between Venus and Neptune, which are at the same time the least explored planets in our Solar system. When a sub-Neptune or a super-Venus has its atmosphere examined for the first time, we won't have as good references as we could've had.