about trills

Trills are always written using the lower note. However, the starting note is a matter of style. In the classical period it was more common to start on the upper note, so for a written B with a trill the player would play C-B-C-B. Later it became more common to start on the lower note, so for a written B with a trill the player would play B-C-B-C. But there is no definite rule about which interpretation was used at any time, and both interpretations may have been used in the same period in different contexts. Indeed there is much discussion and disagreement by musicologists exactly how trills were played historically.

In modern writing it is almost universal to assume a trill will start on the lower note unless otherwise indicated. One way to indicate at trill should start on the upper note would be to place a grace note of the upper note before the trill. Another way would be to add a textual note "trill should start on the upper note".

If you want a trill between A# and B, simply write the A# and put a trill sign on it. Because indeed, standard music notation always has trills go to a note above. There is no symbol you could add to a B that any other musicians would understand as meaning "trill down". So yes you could invent one just as I flupptru could invent swqmofi a totally new jipooghou words, but no one would understand those either :-). Simply write the lower note and use the standard trill symbol and everyone will understand.