Answer for question 1:
According to this paper popular FHE schemes like BGV, NTRU and so on support plaintexts from a ring $R_p = Z_p[X]/\langle f(x) \rangle$ where $n = deg(f)$. The main trick is that the encryption noise should be a multiple of $p$ and the decryption is done by applying the reduction modulo $p$ instead of $2$.
Note: Although the plaintext space is a polynomial ring $R_p$ you can always encode your message $m$ as the free coefficient i.e. $m = x^0 \cdot m + x^1 \cdot 0 + \dots x^{n-1} \cdot 0$. One can argue that this is not really efficient because some coefficients are not used.
Fortunately there are some plaintext packing techniques (see Section 1.1) - a cool way to split the ring $R_p$ in sub-fields $F_p^d$ for which $d$ is a divisor of $n$ (this includes the case $d=1$).
You can view it as operating on contiguous sequences of length $d$ on an array of length $n$ and doing multiple encryptions at the same time.
Answer for question 2:
Additions and multiplications with public constants are (almost) for free in the FHE world, at least for BGV. Take for example the encryption from BGV scheme (page 12) instantiated with RLWE:
Consider $c \in R_q^{2}$ an encryption of $\textbf{m}_1 = (m_1, 0)\in R_q^{2}$. If we multiply the first component $c_1$ with a public constant $\textbf{m}_2 = (m_2, 0)$ and then decrypt using the secret key $s = (1, t_1)$ we get $m_1 \cdot m_2$.
Why? Use the fact that $b = Bt +2e$.
$c \cdot s = c_1 \cdot 1 + c_2 \cdot t = \\
m_1 \cdot m_2 + m_2 \cdot \sum_{i=1}^{N}b_ir_i - t \cdot \sum_{i=1}^{N}B_ir_i = \\
m_1 \cdot m_2 + m_2 \cdot r^t \cdot 2e
$
Now if we reduce the last term mod $R_q$ and then mod $R_2$ we obtain $m_1 \cdot m_2$.
In this way we obtained $Enc(m_1 \cdot m_2)$ starting from $Enc(m_1)$, $m_2$ and added noise by a factor of $m_2$. This is better than encrypting $\textbf{m}_2$ and then do a homomorphic multiplication with a quadratic noise.
Bonus: Addition between $Enc(m_1)$ and $m_2$ can be done with zero noise!