All of the elements that we know about are on the periodic table. Some elements, like the ones with atomic numbers of 93-118, are very unstable. Using the laws of physics (such as the nuclear strong and weak forces), we have calculated that these elements can exist for very short periods of time and in extreme environments, such as really intense radiation or high speeds.

Some of them, like oganesson (atomic number 118), has been man made. It was first made in 2002 in a fusion reaction of lead and krypton. Since then, it has only been made 5 or so times. The element is very unstable and tends to decay back into smaller elements in less than a millisecond. The heaviest naturally occurring element is uranium (atomic number 92). The rest of the heavier elements (with atomic numbers above 92) are too unstable for us to observe in nature, however we can make them in a laboratory using very high energy radiation and fast moving atoms. They do not last long; like oganesson, they only last a fraction of a second before they decay back into smaller elements. In time, scientists may be able to make even heavier elements, in which case we can add onto the periodic table!

Yes, probably. The position of an element on the periodic table depends on the number of protons in the nuclei of that element. Since there are no gaps in the table (that is, nuclei with 1 to 119 protons correspond to known elements), any new elements would be added to the end of the table

Within nuclei, the repulsion between the positively-charged protons is offset by the interaction between protons and neutrons. One can imagine taking an existing element and "pushing in" more protons with enough neutrons to balance the repulsion This is essentially how elements heavier than hydrogen (the first element on the periodic table) are formed , whether naturally or in a lab. However, this process cannot be continued indefinitely:
recent theoretical estimates put the maximum number of nuclides at about 7,000. Note that this is nuclides, not elements. That is, this count includes all isotopes (nuclei with the same number of protons but different numbers of neutrons); the number of possible elements is much smaller.

Still, the number of possible elements from this estimate is greater than the number of known elements, meaning that there are (probably) undiscovered elements.

Yes. The only limit to how many elements there are is the limit by which the atomic nucleus would collapse to become a black hole. This apparently happens at about 2.5 times the mass of the sun. Smaller giant atomic nuclei - called neutron stars - exist.

It is an open question whether there are any stable elements heavier than lead, though.