Response to 1st question: We can define the long exact sequence by compact sets as required. I show the process here.
Take $\{K_i\}$ to be a sequence of compact submanifolds of $M$ such that $M = \bigcup_i K_i$ and $\partial K_i = \partial M \cap K_i$. Then we take the direct limits on first row first to show that
$$H_c^{k}(M, \partial M ; R) \cong \lim_i H^k(M, \partial M \cup \{M\setminus K_i \} \cong \lim_i H^k(K_i, \partial K_i) .$$
We also have
$$H_c^k(M;R) \cong \lim_i H^k(M,M\setminus K_i;R)\cong \lim_i H^k(K,\partial K;R)$$
since $\partial K$ is the collar neighborhood of $M$. In addition, we also have
$$H_c^k(\partial M;R) \cong \lim_iH^k(\partial M, \partial M \setminus \partial K_i;R) $$
since the direct limit of homology is $H_k(M;R)\cong \lim_i H_k(K_i;R)$.
We have the following long exact sequence.
$$
\require{AMScd}\tiny{\begin{CD}
...@>>>H^{k-1}(\partial K_i;R) @>\delta_i>> H^k(K_i,\partial K;R)@>g_i>> H^k(K;R) @>a_i>> H^k(\partial K_i;R) @>y_i>>..
\end{CD}}
$$
As a result, we have the following diagram.
$$
\require{AMScd}\small{\begin{CD}
H^{k-1}(\partial K_i;R) @>\delta_i>> H^k(K_i,\partial K_i;R)\\
@VD_{\partial M}VV @VVD_MV \\
H_{n-k}(\partial K_i;R) @>i_i>> H_{n-k}(K_i;R)
\end{CD}}
$$
The maps $D_M$ and $D_{\partial M}$ are actually isomorphisms by using Poincare duality and Lefchetz duality, since $K$ is compact as well as that isomorphisms commute with boundary and inclusions. Hence this square commutes. It is clear to see that the other squares commute too.
Response to 2nd question: We can't use Lefschetz duality since $M$ is noncompact. However, some of the maps like $D_{\partial M}$ are isomorphisms by Poincare duality for noncompact manifolds. Let's make the sequence into the 5-lemma form and use descending induction to assume 5th map is an isomorphism. Under this assumption, we can simply use 5-lemma to prove that two maps of $D_M$ are isomorphisms and finish the induction.
Let's use descending induction. As we know, when $k \gt n$ all maps are isomorphisms $(H^{0}, H^{-1}, \ldots )$. Let's further assume there exists a number $j$ such that for all $j \gt k$ the maps $H_c^k(M;R) \rightarrow H_{n-k}(M,\partial M;R)$ and $H_c^k(M,\partial M;R) \rightarrow H_{n-k}(M;R)$ are isomorphisms. Assume the statements hold for $k$. Now for the case with $k+1$ we have a 5th isomorphism map $D_{M_A}:H_c^{k+1}(M,\partial M;R) \rightarrow H_{n-k-1}(M;R)$. Let's now expand this map.
Under our assumptions, the last two maps are isomorphisms by the 5-lemma.
$$
\require{AMScd}\tiny{\begin{CD}
H_c^{k-1}(\partial M;R) @>\delta_i>> H_c^k(M,\partial M;R)@>g_i>> H_c^k(M;R) @>a_i>> H_c^k(\partial M;R) @>y_i>>H_c^{k+1}(M,\partial M;R)\\
@VD_{\partial M}VV @VVD_MV @VVD_MV @VD_{\partial M}VV @VVD_{M_A}V \\
H_{n-k}(\partial M;R) @>i_i>> H_{n-k}(M;R) @>>>H_{n-k}(M,\partial M;R)@>>> H_{n-k-1}(\partial M;R)@>>> H_{n-k-1}(M;R)
\end{CD}}$$
As descending induction start from $k \gt n$ to all integers $k$, we have that $D_M$ is an isomorphism.
