Question

predict where an unknown element that has these properties could fit in the periodic table 62 protons and electrons

Answer 1

Position of element in periodic table is depend on the electronic configuration of element.

Element with 62 electrons has following electronic configuration:
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 4f6 6s2

From above electronic configuration, it can be seen that highest value of principal quantum number, where electron is present, is 6. Hence, element belongs to 6th period.

Further, last electron has entered f-orbital, hence it is a f-block element. Position of f-block element is the bottom of periodic table.

Further, there are 6 electrons in f-orbital. Hence, it is the 6th f-block element in 6th period of periodic table.