[QUOTE=SmartMersenne;562211]Our goal is to find better and efficient solutions everywhere[/QUOTE]
As an example, we can try to minimize code length.
In this sense, published solutions are hard to beat.
For base problem, we can save one line by applying the CHAOS trick described in post #23.
After some cleaning, I got 11 lines:

[FONT="Courier New"]10 A = a
20 B = b
30 JMP_ZERO B 90
40 X = A % B
50 A = B
60 B = X
70 JMP_ZERO B 110
80 JMP 40
90 CHAOS 90, 100, 110
100 CHAOS 90, 100, 110
110 RETURN A
[/FONT]
This implementation works for every pair of non-negative integers; if a<b, the first loop from line 40 to line 80 simply swaps A and B.

For bonus problem, there's some trade-off between code length and code speed.
The published solution uses 12 lines and needs 16 steps to reach 1970 paths.
I worked on minimizing the number of steps (keeping length within the 20-line costraint).
Here are my fastest solutions, as I submitted them.
Line 30 can be dropped, by rewriting lines 40 to 90 like in previous base solution.

"BASE-5" SOLUTION: 17 LINES, 7 STEPS.

[FONT="Courier New"] 10 A = a
20 B = b
30 C = A % B
40 JMP_ZERO C 100
50 A = B
60 B = C
70 C = A % B
80 JMP_ZERO C 150
90 JMP 50
100 CHAOS 100, 110, 120, 130, 140
110 CHAOS 100, 110, 120, 130, 140
120 CHAOS 100, 110, 120, 130, 140, 150, 170
130 CHAOS 100, 110, 120, 130, 140, 150, 170
140 CHAOS 100, 110, 120, 130, 140, 150, 170
150 JMP 160
160 CHAOS 160, 170
170 RETURN B[/FONT]

Path counting sequence: 0, 0, 3, 16, 79, 395, 1970

This solution relies on the base-5 representation of 1968 as 30333.
Lines 100 to 140 are used to generate the powers 5^k.
Line 150 "stores" the values 3*5^k, with a one-step delay.
Line 160 "stores" the cumulative sum of such values, with a two-step delay.
Due to line 80, the value at line 150 is increased by 1 at even steps, achieving the +2 cumulative correction needed to reach 1970 at step seven.


"NEAR-BASE-7" SOLUTION: 18 LINES, 6 STEPS

[FONT="Courier New"] 10 A = a
20 B = b
30 C = A % B
40 JMP_ZERO C 100
50 A = B
60 B = C
70 C = A % B
80 JMP_ZERO C 110
90 JMP 50
100 CHAOS 110, 120, 130, 140, 150, 160
110 CHAOS 100, 110, 120, 130, 140, 150, 160
120 CHAOS 100, 110, 120, 130, 140, 150, 160, 170, 180
130 CHAOS 100, 110, 120, 130, 140, 150, 160, 170, 180
140 CHAOS 100, 110, 120, 130, 140, 150, 160, 170, 180
150 CHAOS 100, 110, 120, 130, 140, 150, 160, 170, 180
160 CHAOS 100, 110, 120, 130, 140, 150, 160, 170, 180
170 CHAOS 170, 180
180 RETURN B
[/FONT]
Path counting sequence: 0, 0, 5, 40, 285, 1970.

This solution relies on the representation of 1970 as 5555, using weights 1, 7, 49, and 337 (slightly smaller than 7^3 = 343).
Lines 100 to 160 are used to generate the weighs w(k).
Line 160 "stores" the cumulative sum of the terms 5*w(k), with one-step delay.