1) No, there is no such map. You can only send both of these into the group $K_0(DM_{gm}^{eff})\cong K_0(Chow^{eff})$. The main point (and, possibly, the source of your problems) is that if you want these two maps to be compatible then you should send variety into its motif with compact support! Thus $[A^1]=[P^1]+[pt]\neq 0$! So, the motif of an affine line is "trivial", whereas its motif with compact support is not.

2. Just look at the Mayer-Vietoris triangle $M(G_m)\to M(A^1)\bigoplus M(A^1)\to M(P^1)\to M(G_m)[1]$. Since motives of affine lines are "trivial", the latter morphism yields the isomorphism in question. If you want to study "ordinary motives", look at Hodge structures, yes.

3. If you want to understand the Euler characteristics of (motives with compact support of) a variety then just count points in $V(F_q)$.