Q1 + Q2: At present the Chow motives known to be finite dimensional are precisely those that are contained in the thick tensor subcategory generated by motives of abelian varieties. That is, the motives that can be obtained from motives of abelian varieties by tensorial operations, extensions, quotients and subobjects.

Q3. This is definitely not known at all! Any projective variety is a finite cover of a projective space, and $h(\mathbb P^n)$ is of course finite dimensional.

That said, there is recent dramatic progress in this area. According to a preprint of Ayoub ("Topologie feuilletée et conservativité des réalisations classiques en caractéristique nulle", available on his webpage), the conservativity conjecture in characteristic zero may be within reach (but see the disclaimers in his preprint!), which would in particular imply finite dimensionality for all Chow motives.

Q1 + Q2: At present the Chow motives known to be finite dimensional are precisely those that are contained in the thick tensor subcategory generated by motives of abelian varieties. That is, the motives that can be obtained from motives of abelian varieties by tensorial operations, extensions, quotients and subobjects.

Q3. This is definitely not known at all! Any projective variety is a finite cover of a projective space, and $h(\mathbb P^n)$ is of course finite dimensional.

That said, there is recent dramatic progress in this area. According to a preprint of Ayoub ("Topologie feuilletée et conservativité des réalisations classiques en caractéristique nulle", available on his webpage), the conservativity conjecture in characteristic zero may be within reach (but see the disclaimers in his preprint). This would in particular imply finite dimensionality for all Chow motives which have only even or odd cohomology.

Q1 + Q2: At present the Chow motives known to be finite dimensional are precisely those that are contained in the thick tensor subcategory generated by motives of abelian varieties. That is, the motives that can be obtained from motives of abelian varieties by tensorial operations, extensions, quotients and subobjects.

Q3. This is definitely not known at all! Any projective variety is a finite cover of a projective space, and $h(\mathbb P^n)$ is of course finite dimensional.

That said, there is recent dramatic progress in this area. According to a preprint of Ayoub ("Topologie feuilletée et conservativité des réalisations classiques en caractéristique nulle", available on his webpage), the conservativity conjecture may be within reach (but see the disclaimers in his preprint!), which would in particular imply finite dimensionality for all Chow motives.

Q1 + Q2: At present the Chow motives known to be finite dimensional are precisely those that are contained in the thick tensor subcategory generated by motives of abelian varieties. That is, the motives that can be obtained from motives of abelian varieties by tensorial operations, extensions, quotients and subobjects.

Q3. This is definitely not known at all! Any projective variety is a finite cover of a projective space, and $h(\mathbb P^n)$ is of course finite dimensional.

That said, there is recent dramatic progress in this area. According to a preprint of Ayoub ("Topologie feuilletée et conservativité des réalisations classiques en caractéristique nulle", available on his webpage), the conservativity conjecture in characteristic zero may be within reach (but see the disclaimers in his preprint!), which would in particular imply finite dimensionality for all Chow motives.