Euclidean TSP in NP and square root complexity
Q1. This is a notorious open problem. It is known to be in the fourth level of the counting hierarchy, due to [ABKM]. Not known to be in NP. The problem is not really in computing square roots as claimed in the lecture notes: $n$ bits of a square root of an integer can be computed in time polynomial in $n$ and the bitsize of the integer. The problem is, rather, how close the sum of square roots of integers can get to an integer, without actually being integral.

Q2. The $n=1$ case is of course easy. It's the same as $q \le A^2$, which is in polynomial time, because squaring a rational number is in polynomial time.

Q3. According to this page, the best that's known is that there are integers $a_1, \ldots, a_k, b_1, \ldots, b_k$, all between $1$ and $n$, such that $| \sum_{i = 1}^k \sqrt{a_i} - \sum_{i = 1}^k \sqrt{b_i}| = O(n^{-2k + 3/2})$. This is a lower bound of $\Omega(2k\log n)$ on the number of bits needed to be computed for the potentially harder problem which allows negative coefficients. The best upper bound on the number of bits is exponential in $k$.

Q4. I think the decimal representation could be quite inefficient. The length of the period is the multiplicative order of 10 modulo the denominator, which can be exponential in the number of bits of the denominator.