Is carbon dioxide symmetrical?

Yes

CO2 have 2 double bond, and C have no lone pair
By VSEPR, the 2 bond pair will get as furthest as they can, as there is no lone pair, the double bonds will be at 2 opposite sides
That means the O=C=O bond angle will be 180 degree and the CO2 molecule will be in linear shape.And, as the two C=O bonds are the same(they are of same bond length), so it is symmetrical.

Yes:
point of inversion;
sigma-h plate (plate of symmetry orthogonal to the molecule),
infinite sigma-v plates (plate of sym. cutting along the molecule);
a C-infinite rotationary axis,
infinite C-2 rotat. axes.