Was the Earth's sea-level significantly lower in ancient times?

It's easier to break-down the answers to your questions into parts:

A lot of times, ancient artifacts or even whole ancient civilizations are found buried very deep into the ground surface of the earth.

This may happen occasionally, but I can't personally think of any instance where evidence of a whole civilization, or any significant human artifacts are found buried very deep into the ground (say, below 5m or so) where they have not been deliberately buried, or subject to an explainable event such as flooding or cave-in.

Artifacts from Human history are usually found at the surface, or near to the surface. When human artifacts are buried, it is usually as a result of other human activities such as agriculture, over-building or deliberate burial. Some human artifacts can be covered by flooding laying down sediments, volcanic eruptions or desertification (for example, shifting sand-dunes).

Older artifacts, such as fossils, are often found deeper because their age means that they have been subject to other geological processes that move them around much more within the structure of the Earth.

Entire civilizations that had high buildings have been found many meters below the earth.

See above

Does that mean the earth's sea-level was much lower than it is now? If this is true, then are we experiencing a lower level of gravity?

The sea level on Earth has varied since the last ice-age, but 'sea level' is a difficult context when experienced over millions of years, as over those timescales other geological processes such as tectonic movement (movement of continental plates) uplift and erosion mean that sea-level becomes a 'relative' measure.

Sea level is not related to gravity. The gravity on Earth has always been the same (at least, since animals evolved) because the gravity of the Earth relates to it's mass (how much it weighs).

Sea level is only related to the amount of liquid water on the earth and how that water accumulates across the surface of the Earth from the lowest point up. Sea level does vary very, very slightly due to the effects of different levels of gravity around the world, but these relative differences in gravity are not large and only make a very small difference. The action of the moon in creating tides has a much larger effect.

How can just sedimentation of sand, bury big building to such depths?

Sedimentation of sand tends not to do this, sedements are generally carried by rivers and any large structure under water is generally eroded by the water that brings sediments. On land, there are cases of sand dunes, which are moved by the action of wind, burying structures to some depths. Volcanic ash can also do this, as in the case of Pompeii in Italy.

If lack of habitation by humans eventually results in burying of things, then what about the places like Chernobyl, which has not been inhabited for a long time?

A lack of habitation is not strongly connected with the burying of structures and objects, although the growth of plants around buildings may lead to an increase in biomatter (dead plant material) which will break down into soil and may build up around buildings. Likewise, the erosion of the buildings themselves through the action of rain, wind and ice may cause them to break-up and effectively 'bury themselves' but this would not to be any great depth.

I am not a physics grad or anything, just a curious guy.

Any help is much appreciated.

I'm not sure of the premise of your opening statement:

Most of the times, ancient artifacts or even whole ancient civilizations are found buried very deep into the ground surface of the earth.

In fact, most artifacts are quite close to the surface (less than a few metres) - that's why many of them turn up in ploughed fields or when digging shallow trenches.

It is true that ruins appear to be deeper in major cities (London and Paris), but that's because the old buildings were simply built over. Even then, the Roman walls of London are still only a couple of metres below the surface, and are openly visible to the public from ground floor galleries in some buildings (Greater London Authority building, for one).

There are some odd circumstances - such as the pyramids and sphinx being buried under desert sands. However, desert sands are very mobile and will pile up against things like the pyramids without continuing care.

As for sea level - the coast is constantly changing. There are ancient ports on the English east coast that are now miles inland. There was also once a land bridge, called Doggerland, connecting Britain to Europe tens of thousands of years ago. Sea levels actually rose considerably after the end of the last major ice age, and the polar caps released large volumes of water.

The beginnings of human Civilization by-and-large are coincident with the start of our current interglacial period, known as the Holocene at roughly 10,000BC. At the start of it, worldwide sea levels were about 60m lower than today (and that was down from more than 120 at the glacial maximum 10,000 years prior). They rose quite rapidly after that, to nearly present levels by 6,000 BC.

However, many areas over 40 degrees of latitude were artificially depressed due to the weight of all that ice by about 190m lower than today's levels (yes, it weighed that much). Land moves much slower than water, so these areas are still slowing rising today. So at the higher latitudes you generally see the opposite effect of areas that were previously underwater rising. This caused particularly interesting history for the Baltic, which went through several periods of being a lake, then a sea, then a lake, then a sea, as both water and land levels rose.