If I am right, we can see only those stars that lie within our cosmological horizon, and there may or may not be any stars beyond that. Given last 150 years of using telescopes, and since then our cosmological horizon must have increased by about 150 light years, have we seen any new galaxies, stars? If given enough time, say 500 years, new stars are found, can we be sure they have become visible against not being formed, say, 150 years after big bang?
No. Space expansion doesn't slow down, unlike the expansion of cosmic horizon. That means the cosmic horizon actually swallows galaxies as they travel away from us due to space expansion, reducing their number instead of increasing it. Nevertheless, we still haven't gathered all cosmic radiation from Big Bang that occurred within our current event horizon, so it's still a good while before it begins restricting distance of observable universe for us - currently the speed of light (+space expansion rate) vs age of the universe limits it to a considerably smaller radius.
Yes, we discovered more stars and galaxies, but that is due to improvement of technology, which is still quite far (roughly a third of the way) from observing stars near the Hubble Sphere as the universe formed (ones whose light would be observable today; as opposed to these at the space horizon, whose light will reach us eventually, in infinitely distant future.)
Currently we're still quite a bit away from leaning against the cosmological limits of observability. Currently our limit of observability is the budget for building better devices.
There are a couple of different horizons you should care about. The first is the cosmological horizon, which is the furthest you could possibly see given that photons travel at a finite speed and the universe is not infinitely old. Since nothing travels faster than the speed of light, this is quite literally the furthest we could ever hope to see - everything outside is causally disconnected from us (though this horizon does increase with time, it asymptotically reaches some ultimate distance due to the accelerated expansion of space-time).
Practically speaking, the horizon which we really care about is the cosmic microwave background. This is the point at which the universe became cool enough such the atoms could remain neutral. Why is this the important horizon? Before this, photons simply could not travel very far before interacting with charged particles. Before the surface of last scattering the universe was essentially opaque (it would be like looking through a cloud). Afterwards, however, it was transparent. Everything from the surface of last scattering and later (redshift of $z\sim 1100$ and lower) is said to be part of the observable universe.
Though the universe may still be infinite in size, we will never be able to see these objects (if they exist) if they are passed this horizon. The other thing to remember is that the further you look, the further back in time you're seeing. Galaxies did not always exist. It took time to form larger overdensities like stars, galaxies, and clusters from small perturbations in density during the early universe. After the surface of last scatter, there was a period of time in which the universe was very dark and potentially very boring from an astronomer's point of view. This is the time after atoms became largely neutral yet before the first stars and galaxies "turned on".
Over the past 150 years, telescope technology has improved immensely, so we can now see much further than in the past. Hubble's 'Deep Field' images would not have been possible 30 years ago, and are among the first I know of that show objects near the cosmological horizon.
Also, while the cosmological horizon increases over time, objects near the horizon have a huge redshift (speed) away from us. I don't have the data to prove it, but I suspect objects near the cosmological horizon would disappear past the horizon over time.
As far as observation goes, distance equates to age, so as our technology improves we effectively see farther into the past. The actual increase in distance visible to us is much much greater than 150 light years.
We can actually detect the effects of the big bang itself, in the microwave background radiation of the universe, but sars and galaxies formed much later than 150 years after themig bang (try millions of years :-)