Your $c_n(\lambda)$ is closely related to what Billey, Konvalinka, Swanson 2020 call the aft of a partition: $$ \text{aft}(\lambda) = | \lambda | - \max\{\lambda_1, \lambda'_1\}$$ where $\lambda'$ is the conjugate of $\lambda$, so $\lambda'_1 = \ell(\lambda)$.

Following up on Martin Rubey's comment, your Question 1 is true: For $c_n(\lambda) > n/2$, we have $\lambda \ne \lambda'$ since $\lambda_1 = \lambda'_1$ would make $2\lambda_1 > n$, a contradiction. Therefore, $\lambda$ is not self-conjugate and each partition $\mu \vdash n - \lambda_1$ is counted twice in the coefficient of $t^{\lambda_1}$, once for the partition of $n$ consisting of $\lambda_1$ followed by $\mu$ and once for the conjugate of that partition of $n$.

For your Question 2, a generating function in terms of Gaussian binomial coefficients is given in the related OEIS entry A338621 submitted by Swanson, one of the article authors.

Your $c_n(\lambda)$ is closely related to what Billey, Konvalinka, Swanson 2020 call the aft of a partition: $$ \text{aft}(\lambda) = | \lambda | - \max\{\lambda_1, \lambda'_1\}$$ where $\lambda'$ is the conjugate of $\lambda$, so $\lambda'_1 = \ell(\lambda)$.

Following up on Martin Rubey's comment, your Question 1 is true: For $c_n(\lambda) > n/2$, we have $\lambda \ne \lambda'$ since $\lambda_1 = \lambda'_1$ would make $2\lambda_1 > n$, a contradiction. Therefore, $\lambda$ is not self-conjugate and each partition $\mu \vdash (n - \lambda_1)$ is counted twice in the coefficient of $t^{\lambda_1}$, once for the partition of $n$ consisting of $\lambda_1$ followed by $\mu$ and once for the conjugate of that partition of $n$.

For your Question 2, a generating function in terms of Gaussian binomial coefficients is given in the related OEIS entry A338621 submitted by Swanson, one of the article authors.

Your $c_n(\lambda)$ is closely related to what Billey, Konvalinka, Swanson 2020 call the aft of a partition: $$ \text{aft}(\lambda) = | \lambda | - \max\{\lambda_1, \lambda'_1\}$$ where $\lambda'$ is the conjugate of $\lambda$, so $\lambda'_1 = \ell(\lambda)$.

Following up on Martin Rubey's comment, your Question 1 is true: For $c_n(\lambda) > n/2$, we have $\lambda \ne \lambda'$ since $\lambda_1 = \lambda'_1$ would make $2\lambda_1 > n$, a contradiction. Therefore, $\lambda$ is not self-conjugate and each partition $\mu \vdash n - \lambda_1$ is counted twice in the coefficient of $t^{\lambda_1}$, once for the partition of $n$ consisting of $\lambda_1$ followed by $\mu$ and once for the conjugate of that partition of $n$.

For your Question 2, a generating function in terms of Gaussian binomial coefficients is given in the related OEIS entry A338621.

Your $c_n(\lambda)$ is closely related to what Billey, Konvalinka, Swanson 2020 call the aft of a partition: $$ \text{aft}(\lambda) = | \lambda | - \max\{\lambda_1, \lambda'_1\}$$ where $\lambda'$ is the conjugate of $\lambda$, so $\lambda'_1 = \ell(\lambda)$.

Following up on Martin Rubey's comment, your Question 1 is true: For $c_n(\lambda) > n/2$, we have $\lambda \ne \lambda'$ since $\lambda_1 = \lambda'_1$ would make $2\lambda_1 > n$, a contradiction. Therefore, $\lambda$ is not self-conjugate and each partition $\mu \vdash n - \lambda_1$ is counted twice in the coefficient of $t^{\lambda_1}$, once for the partition of $n$ consisting of $\lambda_1$ followed by $\mu$ and once for the conjugate of that partition of $n$.

For your Question 2, a generating function in terms of Gaussian binomial coefficients is given in the related OEIS entry A338621 submitted by Swanson, one of the article authors.