Let $f$ be a primitive holomorphic cusp form of weight $k$, level $N$ and nebentypus $\chi$, with its $L$-function $L(s,f)=\displaystyle\sum_{n\geq1}\lambda_f(n)n^{-s}$ for $\mathrm{Re}(s)>1$. Let $\alpha, \beta$ be the Satake parameters such that $$L(s, f)=\prod_{p}\left(1-\frac{\alpha(p)}{p^{s}}\right)^{-1}\left(1-\frac{\beta(p)}{p^{s}}\right)^{-1}.$$ One can then define the symmetric square $L$-function to be $$L(s,\mathrm{sym}^2f)=L(s,f\otimes f)L(s,\chi)^{-1}.$$ Such a $L$-function has Euler product $L(s,\mathrm{sym}^2f)=\displaystyle\prod_p L_p(s,\mathrm{sym}^2f)$, and we know that for $p$ not dividing $N$, $$L_p(s,\mathrm{sym}^2f)=\left(1-\frac{\alpha(p)^2}{p^{s}}\right)^{-1}\left(1-\frac{\chi(p)}{p^s}\right)^{-1}\left(1-\frac{\beta(p)^2}{p^s}\right)^{-1}$$ $$=\left(1-\lambda_{f}\left(p^{2}\right) p^{-s}+\lambda_{f}\left(p^{2}\right)\chi(p) p^{-2 s}-\chi(p)^3p^{-3 s}\right)^{-1}.$$
If $N$ is square free, we know that $$L_p(s,\mathrm{sym}^2f)=\left(1-\frac{1}{p^{s+1}}\right)^{-1}$$ and $$L(s,\mathrm{sym}^2f)=L(2s,\chi^2)\sum_{n\geq1}\lambda_f(n^2)n^{-s}.$$
My questions are: Are the two equalities above still true when $N$ is not square free? If it is not:
Can I still write $L(s,\mathrm{sym}^2f)$ in a simple manner using the coefficients $\lambda(n^2)$?
Does $\displaystyle\prod_p L_p(s,\mathrm{sym}^2f)$ still define a $L$-function with $L_p(s,\mathrm{sym}^2f)=\left(1-\frac{1}{p^{s+1}}\right)^{-1}$ for $p|N$?
If I apply the approximate functional equation (Thm 5.3 in Iwaniec-Kowalski's book), do I still get something that looks like $$L\left(s,\mathrm{sym}^2f\right)\ll c(N)\mathop{\sum\sum}_{n,m}\lambda_f(n^2)\chi^2(m)n^{-s}m^{-2s}V_s\left(\frac{nm^2}{XN}\right)+ \text{ something of similar structure },$$ with some $c(N)$ depending only on the primes dividing $N$?
Added: It appears that the answer may be long for full generality. What if we restrict our attention to trivial nebentypus $\chi$?