The cash flow looks like this:
$$AV = 2000\left(\left(1+\frac{i^{(4)}}{4}\right)^{\!40} \!\!\!\! + (0.98)\left(1+\frac{i^{(4)}}{4}\right)^{\!38} \!\!\!\! + (0.98)^2 \left(1+\frac{i^{(4)}}{4}\right)^{\!36} \!\!\!\!+ \cdots + (0.98)^9 \left(1 + \frac{i^{(4)}}{4}\right)^{\!22}\right)$$ where $i^{(4)} = 0.10$ is the nominal rate of interest compounded quarterly.
Explanation: the effective rate of interest per quarter period is simply $i^{(4)}/4$. To account for the payments occurring every other compounding period, we just skip those periods. Because payments are made at the beginning of each half-year, the first payment of $2000$ has had the full $10$ years, or $40$ quarters, to accumulate. To ensure that we have $5$ years of semiannual payments, or a total of $10$ payments, we require that the last payment be reduced by $(0.98)^{10 - 1}$, and that $40 - 2(9) = 22$ is the number of periods that the last payment accumulates interest.
Once you see how this is all put together, the meaning should become plainly obvious. This is why I recommend writing out the cash flow. Actuarial notation comes next. We note that we can write the above as
$$\begin{align}
AV &= 2000(1+j)^{22} \left( (1 + j)^{18} + (0.98) (1+j)^{16} + \cdots + (0.98)^9 (1+j)^0 \right) \\
&= 2000(0.98)^9 (1+j)^{22} \left( \left(\frac{(1+j)^2}{0.98}\right)^{\!9} + \left(\frac{(1+j)^2}{0.98}\right)^{\!8} + \cdots + 1 \right) \\
&= 2000(0.98)^9 (1+j)^{22} \require{enclose}s_{\enclose{actuarial}{10} j'} \\
&= 2000(0.98)^9 (1+j)^{22} \frac{(1+j')^{10} - 1}{j'},
\end{align}$$
where $j = i^{(4)}/4 = 0.025$ is the effective quarterly interest rate, and $$j' = \frac{(1+j)^2}{0.98} - 1 = \frac{113}{1568} \approx 0.072066$$ is the equivalent semiannual effective rate after adjusting for the geometric decrease in payments. It follows that $$AV \approx 40052.28.$$ The claimed answer $40042$ is inaccurate.
Alternatively, using your approach and converting the rate to a semiannual frequency, we have $j = i^{(2)}/2 = 0.050625$ as you stated, and the cash flow is then written
$$AV = 2000 \left((1 + j)^{20} + (0.98)(1 + j)^{19} + \cdots + (0.98)^9(1 + j)^{11}\right) = 2000 (0.98)^9 (1 + j)^{11} \require{enclose}s_{\enclose{actuarial}{10} j'}$$ where now $$j' = \frac{1+j}{0.98} - 1.$$ Either way gives the same result.
As I said in a comment, the amount of the loan should be calculated as $$\require{enclose}
L = 600 a^{(12)}_{\enclose{actuarial}{10} i} = 600\left(1-(1/1.01)^{120}\right)/0.01 = 41820.31
$$
You calculation of the total interest is correct. It's just the total payments less the amount of the loan.
The interest paid in the $10$ payment is $1\%$ of the amount of the loan outstanding after $9$ payments. Since there are then $111$ payments left, you do it it just as you calculated the amount of the loan, but with $111$ in the exponent instead of $120$.
For the amount of principal paid in the $20$ payment, calculate the amount of interest in the payment as above. The amount of principal in the payment is $600$ minus the amount of interest in the payment.
Best Answer
The calculation you show in your work answers a different question, namely: How much should I invest now with a one-time deposit so as to have $\$ 2000$ after six quarters?
For your stated problem, you should use the present value of annuity formula: $V=R\cdot \frac{1-(1+i)^{-n}}{i}$ where $R=\$2000$, $i=.03$, and $n=6$.