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To add to Fedja's comment, the problem of finding a lattice with the smallest possible ratio of covering radius to packing radius is also the subject of a paper of Schurmann and Vallentin (arxiv.org/abs/math/0403272), where they cite a result of Butler for the statement that the optimal ratio is asymptotically 2+o(1)
Cox and Flikkema have some candidate solutions from small n here: doi.org/10.37236/317, and the reference therein will lead you to some literature on the existence and regularity of solutions to this problem (not sure about uniqueness though). In general the arcs of constant curvature and meet in threes at 120º, but not necessarily straight.
You should also be aware that getting a manuscript accepted at a math journal is often a lengthy process and it's quite common to have to wait upwards of six months to hear back from referees. Depending on the timeline for your applications, take into consideration that even in the best case scenario where you are able to write up a novel result, you might only have, say, an arxiv preprint and a journal submission at the time of the application.
One thing to note as you look at the references you cited, is that usually the proportion of discs of different sizes is also perscribed, and I believe the relevant question then is whether the optimal density can always be achieved by a mixture of one or more periodic structures.
This seems possibly related to a previous MO question about irregular, but fair dice. Is your 3D example the same as the example described in this answer? mathoverflow.net/questions/46684/…
One way to describe this configuration is to consider on of the points as the "north pole", then you have a (d-1)-simplex on one lattitudinal cross section (the "tropic of cancer") another one, with the opposite orientation, on the "tropic of capricorn", and another point on the souther pole. The minimum distance is not realized inside each line of lattitude, but only between adjacent ones. Since we have one more point than needed, we can remove the south pole, and pull the tropics south a little bit. This should increase the minimal distance slightly.
For unit squares I believe you should be able to use the structure in the mathworld illustration, replacing the circles with alternating coordinate-aligned squares and 45-degree-rotated squares so that each square of one orientation is wedged between three squares of the other orientation.
Why do you write "probably" the thinnest? The packing by Böröczky cited on that page and in the linked question has density zero. Are you suggesting there may be ones with negative density?
