The proportion of heads after the first ten tosses is zero because the first ten are all tails. The proportion of heads after the first hundred tosses is
$$
{45\over100}=0.45
$$
Similarly for 3 and 4, you get $0.495$ and $0.4995$.
The question is asking you to calculate the numbers rather than say what the probability of heads or tails is.
It sounds like you already have the intuition since you understand that the answer is obtained by dividing the number of outcomes with exactly 3 heads by the total number of outcomes. From here it's a matter of understanding how to calculate these two things.
The total number of outcomes is simply $2^6 = 64$ since we're tossing a coin 6 times and each toss has only two possible outcomes.
The number of outcomes with exactly 3 heads is given by ${6 \choose 3}$ because we essentially want to know how many different ways we can take exactly 3 things from a total of 6 things. The value of this is 20.
So the answer is $20/64 = 5/16$.
The error you made is thinking that "number of outcomes with exactly 3 heads" is equal to "half of the total number of outcomes of 6 tosses." If this were the case then logically, "exactly 3 tails" must also be exactly half of the total outcomes. This means that "exactly 3 heads or exactly 3 tails" must describe all possible outcomes (because each scenario joined by the "or" would have probability $1/2$) but this is clearly not the case since we can have, e.g., 1 heads and 5 tails, etc.
To put it another way... You said any sequence is equally likely. That is correct. But sequences containing exactly 3 heads do not make up half of the total number of sequences. Therefore it does not follow that the probability is $1/2$. The easiest way to see this clearly is to list every possible outcome. But for 64 outcomes it can be tedious, so let's do it with a simpler and similar problem.
Say we want to know the probability of getting exactly 2 heads if we flip a coin 4 times. Unless I'm misunderstanding your misunderstanding, your earlier thinking would lead you to believe the answer is $1/2$. But if we list all possible outcomes:
HHHH
HHHT
HHTH
HHTT
HTHH
HTHT
HTTH
HTTT
THHH
THHT
THTH
THTT
TTHH
TTHT
TTTH
TTTT
We see that only 6 of them have exactly 2 heads. $6/16 = 3/8$. And if we do this problem the way I answered the original one, then the total number of outcomes is $2^4 = 16$ and the total number of outcomes with exactly 2 heads is ${4 \choose 2} = 6$. So we again get $6/16 = 3/8$.
Best Answer
Exactly 4 heads: This boils down to choosing 4 "spots" out of the 12 "spots" to put the 4 heads outcomes. That is, $\binom{12}{4}$.
At least 2 heads: $2^{12}- \binom{12}{0} - \binom{12}{1}$ subtracting the possibility of 0 or 1 heads.
At most 8 heads: $2^{12}- \binom{12}{12} - \binom{12}{11} - \binom{12}{10} - \binom{12}{9}$ subtracting the possibility of 9 or more heads.