The answer you link to says the minute hand gains $55$ minutes on the hour hand, not the other way around. As you say, in one hour the minute hand moves $360^\circ$, which the answer calls $60$ minutes. In one hour, the hour hand moves $30^\circ$, so the minute hand gains $330^\circ=55$ minutes.
Your answer, (b), is correct, but the following may help clean up any problems with your reasoning:
You want to find out how many hours, $t$, elapse before the wall clock, $W_c$, and the table clock, $T_c$, show the same time again.
Since the wall clock gains $2$ minutes every $12$ hours, it really gains $\frac{t}{360}$ hours every $t$ hours (e.g., $\frac{12}{360}=\frac{1}{30}$ and $\frac{1}{30}$ hours is equivalent to $2$ minutes). Thus, we can let the total time elapsed on $W_c$ for $t$ hours be
$$
W_c = t+\frac{t}{360}.\tag{1}
$$
For the table clock, $T_c$, we actually lose $2$ minutes every $36$ hours; that is, we lose $\frac{t}{1080}$ hours every $t$ hours (for example, $\frac{36}{1080}=\frac{1}{30}$ and $\frac{1}{30}$ hours is equivalent to $2$ minutes, as described above). Thus, we can let the total time elapsed on $T_c$ for $t$ hours be
$$
T_c = t-\frac{t}{1080}.\tag{2}
$$
Now what? This depends on the kind of clock you are using. If you are using a regular wall-clock that does not differentiate between AM and PM (i.e., a 12-hour clock), then we will need to figure out when
$$
W_c-T_c=12.\tag{3}
$$
However, if you are using a clock that does differentiate between AM and PM, then you will need to figure out when
$$
W_c-T_c=24.\tag{4}
$$
Using a 12-hour clock: We substitute $(1)$ and $(2)$ into $(3)$ to get
$$
\frac{t}{360}+\frac{t}{1080}=12\Longleftrightarrow 4t=12960\Longleftrightarrow \color{red}{t=3240}.
$$
Thus, $3240$ hours will have elapsed. Note that
$$
\underbrace{3240}_{\text{hours}} = \underbrace{19}_{\text{weeks}}\cdot \underbrace{168}_{\text{hours/week}} + \underbrace{48}_{\text{hours}}.
$$
Thus, $19$ weeks and $48$ hours will have elapsed. Since your clocks began on a Tuesday at noon, they will next meet again on $\boxed{\color{red}{\text{Thursday at noon}}}$.
Using a 24-hour clock: We substitute $(1)$ and $(2)$ into $(4)$ to get
$$
\frac{t}{360}+\frac{t}{1080}=24\Longleftrightarrow 4t=25920\Longleftrightarrow \color{red}{t=6480}.
$$
Thus, $6480$ hours will have elapsed. Note that
$$
\underbrace{6480}_{\text{hours}} = \underbrace{38}_{\text{weeks}}\cdot \underbrace{168}_{\text{hours/week}} + \underbrace{96}_{\text{hours}}.
$$
Thus, $38$ weeks and $96$ hours (four days) will have elapsed. Since your clocks began on a Tuesday at noon, they will next meet again on $\boxed{\color{red}{\text{Saturday at noon}}}$.
Best Answer
The "gains or loses" question is fairly straightforward: a normal clock takes $60\cdot \frac {12}{11}=65\frac 5{11}$ minutes for the minute hand to catch up to the hour hand, while our faulty clock takes $66$ minutes which is $\frac 6{11}$ minutes longer. This indicates that the faulty minute hand is moving slower than the normal minute hand, which means that the faulty clock is "losing" time.
How much time is lost is then $\frac 6{11}$ minutes in $66$ minutes, which is the solution $2$ calculation you have already done, but does not correspond to $6\over 11$ minutes per hour, which would result in $2\frac3{11}$ minutes lost over $4$ hours.