I am visiting Greece just now, and was amazed that so many ruins have e.g. columns where part is clearly a modern reconstruction for structural reasons. But also many statues and delicately-detailed stoneworks are removed to a museum and a replica is on site, and yet other stones we are allowed to walk on and touch.
Many of these sites were basically heaps of stone blocks for two thousand years, which aren't as impressive to visit as, e.g., the Stoa of Attalos in the Athens Agora which was built in the 1950s on top of, and containing parts of, the original building.
And yet. The decision was clearly made at somepoint by someone to STOP rebuilding, as at most sites only a few columns or an arch are all that stands. How is that line drawn, and by whom?
In relation to authenticity, the reconstruction of archaeological remains or historic buildings or districts is justifiable only in exceptional circumstances. Reconstruction is acceptable only on the basis of complete and detailed documentation and to no extent on conjecture.
This is from UNESCO's Operational Guidelines for the Implementation of the World Heritage Convention, essentially the sets of principles that guide the custodianship for sites inscribed the World Heritage list. UNESCO designation is a very rare and prestigious honor for an archaeological site so this is not what legally governs most sites, but I think it serves as a useful statement of the highest international principles. It also is useful because it shows how subjective those principles can be--the idea that any changes must be documented, identifiable, and reversible are basic standards, but "exceptional circumstances" is an exception big enough to relocate the entire Abu Simbel and Philae temple complexes onto. Said relocation gets to some of the tension behind the UNESCO World Heritage project, which purports to show sites frozen in amber but necessarily acts as an agent of profound change. UNESCO itself has put out a nice little blog post on its changing attitudes towards this topic, and how it attempts to navigate the tricky questions involved, which I think does a great job of answering some of your questions.
But that is of course UNESCO and questions of principle, in terms of actual practicalities it is substantially--not exclusively, but substantially--about funding. I do not know how much you are going to be traveling around Greece, but it is probably going to be hard to not see massive disparities in how well sites are preserved and reconstructed based on funding and tourist interest. At one end of course is the Acropolis, which is perpetually in a state of truly heroically painstaking restoration, and on the other is something like the nearby Amphiareion of Oropos which is essentially unmarked and (at least around 2010 when I was there) unfenced off. Even within Athens the excellent Piraeus Museum is by no means neglected but is a far cry from the glitzy and hyper modern Acropolis Museum.
What this all means is that while there are certain hard practical issues informing reconstruction (safety, how much conjecture is needed, the feasibility of reconstruction versus a nearby replica, etc) the fundamental question is political. The site is not owned by archaeologists, it is owned by the local society, acting through its government. Archaeologists and grant giving institutions are guests that act in an advisory role, but the actual question of what to reconstruct is not up to them.
On an anecdotal level, I was on a site which had some substantial reconstruction on it (on the order of the rebuilding of the Stoa of Attalos) and the director had the position that while he would have personally done some things different at the end of the day it was great for the site.
As a side note the Acropolis of Athens is like a living museum of different theories of reconstruction. From the (lamentable, in my view) clearing of practically all post-Classical remains to the early twentieth century reconstructions that did terrible damage to the marble to the modern block-by-block "unexplosion" of the Parthenon it has been the testing ground of every theory and practice.