How to encode shortest brainf_ck strings
I can give question number 1. a go:

++++++++++


The first series of +'s defines first memory cell (cell0) to be a loop counter to 10. 

[&gt;+++++++&gt;++++++++++&gt;+++&gt;+&lt;&lt;&lt;&lt;-]


Above is the loop that executes 10 times. The loop adds 7 to cell1, 10 to cell2, 3 to cell3 and 1 to cell4 for each iteration. After 10 iterations we have the following in the first 5 memory cells: 0, 70, 100, 21, 10

&gt;++.&gt;+.+++++++..+++.&gt;++.&lt;&lt;+++++++++++++++.&gt;.+++.------.--------.&gt;+.&gt;.


Above is kind of self explanatory, it just moves the memory pointer to the value that is closest to what is needed, does some minor inc's and dec's and outputs.