Is the existence of chirality center equivalent to chiral?
The answer to both questions is yes.

In regards to the first question, meso compounds like tartaric acid have multiple chiral centers, but are not chiral overall because they have a mirror plane. 

For the second question, a compound can be geometrically chiral without having a chiral center. A cool example class of compounds is helicenes, which are helical arrangements of aromatic rings. These compounds have axial chirality, where the stereoisomers are the left and right handed versions of the helix.