Could well be your Minisail is a Sprint I. These came with the black spars, including the "non- bendy boom", had usally six cleats for rigging control and were a bit stiffer built. Hard to tell in words how much stiffer, but if the flat mid to aft bottom of the hull doesn't flop in choppy conditions, chances are it's a Sprint I.
A good way to rig is illustrated in Rupert's blog. Here
Some more ideas on this subject were in the Minisail yearbook 2012
(also written by Rupert).
In the above example the rowlock is prevented from sliding down by the thickening element on the mast.
A different way to do this is like Rob discribed above, or to mount a string with a loop over the bottleneck of the mast and a second loop on the other end to hold the boom (works fine but you will loose the string during de-rigging one day as I found out, bummer).
Yes it's better to have a bung on the bottom of the mast. Mind there are no specific aftermarket parts for Minisails.
So you might consider to make a DIY mast bottom bung, top and bottom if both are missing. The bottom bung will prevent the mast biting the bottom of the hull, the top bung will prevent the mast filling up in case of a capsize.
Some have succesfully re-shaped Laser bungs, I make mine as shown in this illustration
If it would turn out the top section of your masts were both shortened (they could have been shortened for example to make them Topper sail compatible, maybe not a bad thing to have a 13 year old starting in a MS), you could insert a wooden batten to re-extend them. Jam some foam in the gaps, again, to prevent the mast filling up in case of a capsize.
Maybe replace the 40 year old waterplug in the transom to make it watertight. The one present does leak without a doubt.
About the toe straps; On a certain moment be brave and tug then merciless. If the fasteners give way, replace all 12 of them.
The missing king-pin for your rudder; It's a simple rond sectioned rod really. The silly matter is that it's a hard to find diameter in standard stock. Tom (our MS encyclopedia) found out some screwdriver shafts are perfect to do the job: stainless cheap and available.
Consider this solution before drilling the gudgeon plates to size of a new rod; Your setup would not match any other Minisail gear anymore.