Why physical / cosmological models that have been falsified weren't abandoned? Before anything, I'd like to say that I'm a layman (non physicist) and english is not my main language; I apologize if my choice of words make me sound rude / arrogant, that is not my intent.
I often watch high level (abstract) Youtube videos (exclusively from PBS Space Time)
and read Wikipedia pages about cosmology,
and I fail to understand why some models have not been abandoned.
Our models predict that galaxies should have X density,
but our observations say that they have Y density.
Instead of considering the model falsified,
we proposed an almost undetectable and basically unfalsifiable "thing";
dark matter.
Our models predict that the universe should have X age or Y radius,
but our observations don't agree with it; we proposed inflation.
Our models predict that the universe should (or shouldn't)
be expanding at X rate,
but our observations suggest otherwise.
We propose another almost undetectable and unfalsifiable "thing",
dark energy.
Our models predict that the edge of galaxies should spin with X velocity,
but our observations suggest other values;
we propose dark halos.
Our models predict that galaxies should move in such and such directions,
but our observations suggest there is such thing as dark flow.
I'm fully aware that I'm probably mixing different models, theories and areas of knowledge here, but as a layman I put all those artifacts in the "what is going on with cosmology bucket?".
So, first, why those mismatches between predictions and observations are not enough to consider such models falsified?
Second, can someone please explain to me, in simple words, why are we doing this? Patching theories with "dark stuff", instead of proposing or studying new theories?
I read somewhere about theories that can explain some of these phenomena without invoking "dark stuff", like the "janus model"... But since I'm not mathematically proficient, nor french speaking, I couldn't read anything about it.
 A: We should start by saying that just because we use the collective term physicists this does not imply any unanimity in their views. For all the points you mention you'll find disagreements within the physics community, and from physicists in the mainstream not just from those on the fringe.
Still, if we take for example dark matter I think it's fair to say that the existence of dark matter is the majority view despite a lot of work from the MOND community to find alternate explanations, and it's fair to ask why this is.
The answer is that physicists generally believe there is a simple explanation for everything, where by simple we mean a theory based on a few simple and clearly defined principles. Right now our leading theory for explaining gravity is general relativity, and despite its fearsome reputation GR is based on just a few remarkably simple principles. Physicists talk about beauty in physics, and while different people mean different things by this I suspect most of us would agree that GR is a beautiful theory. And GR works. Whenever we've been able to test it directly we find the theory correctly predicts what we observe.
The problem is that galaxy rotation curves and galaxy cluster dynamics do not match the predictions from GR and there are only two plausible explanations for this:


*

*GR is wrong

*our observations are wrong
And as you've noted the mainstream view is that the observations are wrong, and specifically that the observed matter density is not the true density because there is dark matter present that we can't see. So your question comes down to why we believe the observations are wrong rather than GR is wrong.
And there are multiple reasons for this. I've already mentioned that GR is beautiful (whatever that means) and that where we can test it directly we find it to be correct. But also it turns out to be difficult to modify GR without breaking it. The very simplicity of GR means it is hard to find other theories that give the same results in our experimental tests but also explain the galaxy rotation and cluster dynamics problems. There are other theories like the Brans-Dicke and TeVeS theories but these look clunky and ill motivated next to GR, and even they struggle to explain all the effects we attribute to dark matter.
So modifying GR turns out to be awkward and clumsy. By contrast adding dark matter turns out to be a straightforward way of explaining all the observations. The bottom line is that adding dark matter is far simpler than modifying GR, and that's why it has emerged as the mainstream view. But I must emphasise that GR is not a religion that physicists treat as dogma. There has been and remains lots of physicists proposing alternatives to dark matter - it's just that so far none of the ideas have been convincing.
I have focused on the dark matter issue because it is the most studied area. You also mention dark energy, and I would guess this is a far more divisive issue. The popular science media may give the impression that dark energy is an established fact, but most of us regard it as just as a working hypothesis. Then when we come to inflation you'll find prominent scientists publishing articles rejecting it.
A: "why those mismatches between predictions and observations are not enough to consider such models falsified?"
Well, the unmodified models are falsified because they made wrong predictions, but the modified models make right predictions so they are not yet falsified. 
In a sense you're asking, why do people modify wrong theories so they are right.. 
"why are we doing this? Patching theories with "dark stuff", instead of proposing or studying new theories?"
Two reasons. First, it's not so easy to make a whole new theory unrelated to what came before. Second, the old theories are logically connected to a lot of things that do otherwise work, like Einstein's theory of gravity. And maybe we can add a third reason, which is that it's easy to add things to these theories (new fields, etc), so it's natural to consider modifying them rather than just chucking them out. 
A few slightly more concrete comments: 
Inflation is slightly different to the other cases, and it's not introduced for the reasons you say (age or size of universe wrong). It's there mostly to explain why the universe is homogeneous on large scales. But also, inflation is a generic kind of behavior in a universe governed by general relativity. If you have a scalar field that starts away from its minimum energy value and moves slowly towards it, you get inflation. Usually one supposes that there is a whole new field, the inflaton, which caused this to happen, but there is some possibility (depending, I guess, on how quantum gravity works) that the Higgs could be the inflaton. 
This might be a good moment to mention another reason why modern physicists feel free to postulate new kinds of matter or fields, in order to explain astronomical mysteries. The reason is that unified theories motivated solely by particle physics reasoning, are full of new particles and fields too. In the particle physics context, these extra types of particles generally arise because one posits large "symmetries" which are partly satisfied by the known particles, and which therefore require new particles to fill out the symmetry. 
Among these new particles, one often finds candidates to play the role of inflaton or dark matter, and in fact - since these particle theories have to be consistent with astronomical data as well as with earthbound experimental data - one would have to calculate the properties of a cosmos filled with these new kinds of matter, just in order to test them against observation, even if we didn't already have known astronomical anomalies providing additional motivation. 
