I'm sure this will vary by individual, but for me:

70+21

(40 * 25) * 3 + 43 + 3 * 75 + 3

(123 * 40) + 123 * 2

paper and/or calculator for the last

sin(PI/3) is an interesting case. I don't deal with trigonometry very often. If the problem is in the context of a project where I will be using trigonometry frequently, I'll go refresh my memory about the easier-to-retain points in addition to what already I remember: sin(N * PI) = 0, sin((2N-1/2) = -1 and sin((2N+1/2) * PI) = 1 for integer N; given a unit circle and an angle between the line segment [(0,0), (0,1)] and another line segment reaching the circumference of that circle, sin(angle) = y, cos(angle) = x of that point (x,y) on the circle (this latter used as a mental-vision sanity check on any answer proposed). If it's truly a one-off, I'll just use the calculator.

For me basic maths ends with

    (3.1^2 - 3^2)/0.1 = (9.61-9)/0.1 = 0.61/0.1 = 6.1

    (3.01^2 - 3^2)/0.01 = (9.0601-9)/0.01 = 0.0601/0.01 = 6.01

    (3.001^2 - 3^2)/0.001 = (9.006001-9)/0.001 = 0.006001/0.001 = 6.001

When you get to that point you are ready to start calculus and see that d/dx x^2 = 2x and that the derivative of x squared at 3 is 6.

There is a smooth transition from arithmetic to calculus with the laws of calculus having to be what they are or else approximations, such as delta-x = 0.001 are not going to approximate them.

I would need paper for all but the first, and could not solve the 4th without a calculator. Not sure what the norm is, here's my data point.

More of a curious question but are you saying you're incapable of solving the last one without paper or just that its really not worth the time. Any of those example questions I could do without paper (I'm 23 for a reference) but I certainly wouldn't attempt the last one without it unless someone was challenging me since it'd take more time than what its worth.

With the problem in front of me, I would not think doing the last one on paper would speed up things. On paper, it would feel like

    Load X
    Triple
    Store Temp
    Load Temp
    Load Y
    Add

I would optimize that store/load away. If, on the other hand, someone told me the problem, I probably would not remember Y by the time I needed it (seven plus or minus two is about two, apparently)

pi/3 is 60°, and the sine of that is √3 / 2. But yes, that's not something one calculates by head, that's memorization. And working out √3 / 2 into decimals also takes a calculator, though I can approximate by knowing the first two significant digits would be 1.7 / 2 = 0.85.

One can memorize trig tables like multiplication tables, and I do find that useful from time to time. I remember sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°, and how to flip the signs to get all four quadrants from 0° to 360°.