My 2 year-old cat is displaying some strange peeing behaviors.  
First of all, she will not pee when I am away from home.  Many times
it is 12 hours or more before she pees (I've been keeping track) and
she will not go until after I come home from work.  When I am home, so
usually pees about every 6-7 hours, sometimes more frequently.
Secondly, she has the habit of frequently peeing over the side of the
box.  She will start out squatting normally, then as she is peeing she
will raise her butt up, causing the stream to go up the side of the
box or over the edge.  At other times she pees normally.  She seems to
do this butt-raising thing more frequently when she hasn't peed for a
long time. Thirdly, several times I have seen her squat and appear to
pee, but nothing happens.  She doesn't seem to be straining or in
pain.  She also sometimes smells the box, but won't go in.  She was
peeing on a throw rug a couple of months ago, but so far she has not
peed anywhere else.  I have had her urine analyzed twice and taken her
to the vet for a physical exam, but they can find no medical reason
for these strange behaviors.

Things I've done: I've tried 5 different kinds of litter—scented and
unscented, several sizes and types of boxes and tried them in
different locations. I keep buying deeper boxes, but it seems the
deeper the box, the higher she pees.  I've tried a covered box, but
she wouldn't go in it.  At present I have 2 litter boxes in the
bathroom—one in the bathtub, which is the main one she uses
(thankfully!).  I use two types of flushable litter.  I keep the boxes
scrupulously clean.  I clean the errant pee with a mild soap and/or
Nature's Miracle.

She is a 2 year-old Russian Blue.  She is spayed but not declawed.  I
adopted her 6 months ago.  She is an only cat but used to live in a
house with another cat.

Any ideas what could be causing these strange behaviors and what I can
do?

Thanks,

Holly