There is a survey [article][1] 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*][2] by Berndt (Springer, 1991, pp. 244-246). The problem  stems from Ramanujan's work on modular equations of degree 3... 


  [1]: https://www.google.com/books?hl=en&lr=&id=TT1T8A94xNcC&oi=fnd&pg=PA215&dq=related:DJmDgKBpMq0J:scholar.google.com/&ots=ujqGzfahmA&sig=FpslhlMUpn6dFwItc5FINa7XYE0#v=onepage&q&f=false
  [2]: https://books.google.com/books?id=wLJ-Dtj193MC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false