What the text says
There is an English translation of The Aryabhatiya with notes is available online.
Regarding the spherical Earth, the text says (page 64 of the book)
The sphere of he Earth, being quite round, situated in the center of
space, in the middle of the circle of asterisms, surrounded by the
orbits of the planets, consists of water, earth, fire, and air.
Regarding the second claim, the text says (page 15 of the book)
A yojana consists of 8000 times a nr. The diameter of the Earth is 1050
A nr is the height of man. Assuming the men of Aryabhata's time were a bit shorter than at the present time, let us suppose this to be 1.6 meters. This gives one yojana as as 12800 meters. 1050 yojanas is then 13440 km. The actual diameter of the Earth is 12742 km.
How accurate is this translation?
The translation I am using is a scholarly work published in 1930 by Walter E. Clark, a Professor of Sanskrit at University of Chicago. This translation was not made from original manuscripts, but from earlier published translations which were cross referenced with abstracts of Aryabhata's work by later Indian authors. The original manuscripts are listed in the preface of in the link; the later Indian commentators include Varahamihira's siddhantas, the Suryasiddhanta, Lalla's Sisyadhivrddhida, and Bramagupta's Brahmasphutasiddhanta and Khandakhadyaka.
For the first claim of a spherical Earth, Lalla's work summarizes this claim, attributing it to Aryabhata. It is also included by two other ancient writers.
For the second claim of the diameter of the Earth, both Brahmagupta and Lalla include the same calculation.
I searched for original manuscripts of The Aryabhatiya, but wasn't able to find anything. I don't read Sanskrit anyways, so I doubt finding an original manuscript would do me much good. However, since both claims are published in a scholarly work, and backed up by other ancient sources, both claims appear to be true.