You remember incorrectly. Calculus was found by Archimedes, G. Saint-Vincent, Galileo, Kepler, Descartes, Pascal, Cavalieri, Fermat, Barrow,
Wallis, Brounker, Huygens, Leibniz, J. Gregory, N. Mercator, Newton, Cotes, Taylor, Torricelli, Bernoulli brothers, to name only the most famous ones.
As every big enterprise, this was a collective enterprise.
The problems which led to its development are: finding areas and volumes (integration), finding tangents to curves (differentiation), finding maxima and minima of functions and functionals (calculus of variations) and expansion of functions into power series which was used for solving differential equations arising in geometry and physics.
But if by "calculus" you only mean the differentiation rules and Newton-Leibniz formula, these were found by Newton and Leibniz, independently.
But this is just one theorem of calculus.
To answer your second question, yes, Newton (and Leibniz and Bernoulli) also knew integration and differential equations. Integration was developed by Eudoxus and Archimedes, and this is the oldest part of calculus. Differentiation as a tool of finding extrema was also used by Archimedes
(and by Fermat, and by others).
Ref. N. Bourbaki, Elements of the history of mathematics.
Remark. Since my mentioning of Archimedes triggered so many comments, let me cite Nicolas Bourbaki, the essay on History of Calculus (my own translation):
The greatest mathematical discovery of the Greeks was their method of treatment of problems which we call integral calculus. Eudoxus gave the first examples of application of this method when he determined the volumes of a cone and a pyramid; this reached us in more or less adequate description by Euclid (VII, Prop. 7, 10). But most importantly, almost all works of Archimedes are devoted to these problems, because of an exceptional luck we can read them in the originals, in his beautiful Doric dialect..
He also mentions that Archimedes was by far the most cited mathematician in
Let me add that all surviving works of Archimedes are easily available in English translation to which I send all those who have any doubts about who invented integration. And many commentaries to them are also available. But for a short and non-technical history of calculus in 17th century (and the role of the Greek heritage in it) I recommend the article of Bourbaki cited above.
BTW, Newton himself described his main contribution to calculus as:
One can solve any differential equation by plugging a power series with indetermined coefficients to it, and find the coefficients one-by one.
(I slightly modernized his language). This is not taught in modern elementary courses.