The Ramanujan Problems.

There is a survey article by Berndt, Choi, and Kang devoted to the set of 58 Ramanujan's problems. They indicate that the questions had originally appeared in the problems section of the Journal and apparently the editors published readers' solutions in subsequent issues.

Concerning your question 1, let me just quote from the Introduction to the survey:

Several of the problems are elementary and can be attacked with a background of only high school mathematics. For others, significant amounts of hard analysis are necessary to effect solutions, and a few problems have not been completely solved.

An elementary solution to the specific geometric problem you've mentioned can be found in Ramanujan's Notebooks, Part III by Berndt (Springer, 1991, pp. 244-246). The problem stems from Ramanujan's work on modular equations of degree 3...

Concerning the planimetry problem: Having a suitable background, it is not hard to produce many puzzles like this. For example, here is one more pair of points on that mysterious circle: intersect a circle centered at $B$ with radius $BM$ and a circle centered at $A$ with radius $AB$, then the two intersection points are also on that 8-points circle.

This circle is a circle of Apollonius with foci at $A$ and $B$. More precisely, it is the locus of points $X$ such that $BX:AX=\sin\alpha$ where $\alpha=\angle BAM=\frac12\angle BAC$. The fact that this is a circle orthogonal to the original one is a general property of circles of Apollonius, and verifying that the ratio of distances equals $\sin\alpha$ for each point is straightforward.

An unelegant solution (to the described problem) which works in principle: Transform everything in algebraic equations, eg. by setting $A=(-1,0),B=(1,0),C=(x_C,y_C),M=(x_M,y_M),\dots$, $x_C^2+y_C-1=0,x_M=1/2(1+x_C),\dots$ and compute a Groebner basis. Doing a few numerical examples proves then that there is at least a large enough set of real solutions.

This is of course ugly but has the advantage that you can give the heavy work to a machine.