The figure to think about consists of a "triangle" of three circles of radius 1, all tangent to each other, and their circumcircle, which has a radius of $1+\frac{2}{3}\sqrt{3}\approx 2.15$.
If we increase the size of the outer circle by epsilon, then we can separate the inner circles a little bit from each other so that we can fit some paths inbetween them, and between them and the outer circle.
Call the inner circles A, B, and C, and call the outer circle D.
The idea is that a long flowing curve can be drawn inside D by wrapping one end around A, then winding the two strands repeatedly around A and B (but staying outside C), and then finally, to end the curve, sending the two strands in opposite directions around C so that they can meet and the curve is complete.
This construction answers question #2, as well as the intent of question #3, which I think is only interesting if it asks for "a number $l>2\pi$". We can see that many lengths $l$ will not be possible, since the repeated loops around A and B must occur a [nonnegative] integer number of times. (Note that one strand could loop once more than the other.) Of course, the slight wiggle room on each loop as you go from A to B (e.g. straight or curved) means that after some number of loops (independent of $\epsilon$), every length is possible. That would make a nice question #4.
The point of question #1 (and to some extent #3) is to prove that this construction is the best possible.
This is of course harder than simply finding the construction.
To do this, we first note that there must be many "roughly parallel" curve parts very close to each other somewhere. (A "bundle" of flowing curve parts.) Although it may seem intuitively obvious, we can prove it by arguing that there are a lot of curve parts on a long curve, and a small area (which there are only a fixed number of inside D) cannot support two curve parts that are going in significantly different directions.
Next, we consider the region inside the curve. This region is very thin where it appears inside a bundle, but any thin part of the region can be followed (in either direction) until it becomes big.
In fact, it must become big enough to fit a circle of radius 1 inside it.
To see this, consider the sub-skeleton
formed only by maximal disks of radius less than one.
This sub-skeleton cannot approach the boundary perpendicularly, and its endpoints are locations where a disk of radius 1 can fit entirely inside the curve. Since this sub-skeleton must exist inside a bundle, it must have at least two endpoints, supporting two disjoint disks.
Similarly, we can consider the sub-skeleton of the region outside the flowing curve. Again, it must be possible to place disks of radius one at the endpoints of this sub-skeleton. Only one of these disks can lie partly outside D, since if they both did then the flowing curve inside D would not be connected. So at least one of the disks must be fully inside D.
This gives us a total of three disjoint disks of radius 1 that must lie inside D, proving that the radius of D must be greater than $1 + \frac{2}{3}\sqrt{3}$ if it encloses a long flowing curve.
