For any manifold (compact or not), compactly supported cohomology is Poincare dual to  (ordinary) homology, via capping with the fundamental class, which is an infinite chain (i.e the sum of all the top simplices in a triangulation).  Likewise, (ordinary) cohomology is Poincare dual to homology with locally finite infinite chains. In notation, $ H^{n-k}_{comp}(X)\cong H_{k}(X)$ and $H^{n-k}(X)\cong H_{k, inf}(X) $. 

  In particular the answer to question 2 is no, $H^{k}_{comp}(X)\not\cong (H_{k}(X))^* $ (I assume you mean  field coefficients), but rather H^{n-k}_{comp}(X)\cong H_k(X)$. It just so happens when $n=2$ and $k=1$, e.g the annulus,  and field coefs, that $ H^1_{comp}(X)\cong H_1(X)\cong (H_1(X))^*$ (the first iso PD, the second iso not natural, depends on $H_1(X)$ f.g.)