All of the problems you describe can be seen in MS and probably can be worsened with a weakened pelvic floor.
The urge incontinence and the incomplete emptying are the two most common urinary problems in MS. The leaking without any sensation of needing to urinate is more likely also an MS problem.
A weak pelvic floor typically causes stress incontinence and much less often urge incontinence.
I think it is important to understand how we hold our urine successfully and what keeps it from leaking out along with understanding how we empty our bladder successfully. I wrote an HP called "The Neurology of Bladder Incontinence". In this I describe the balance between the strength of the urethral sphincter and the pressure inside the bladder. If the sphincter pressure is higher than the bladder pressure the urine will stay in the bladder. If the pressure inside the bladder rises higher than the tightness of the schincter muscle then urine will escape.
http://www.medhelp.org/health_pages/Multiple%20Sclerosis/The-Neurology-of-Bladder-Incontinence/show/758?cid=36
Your problem of spontaneously leaking urine is most likely neurological, I think. It could be from inappropriate relaxation of the urethral sphincter or it might be from an unsensed rise in pressure within the bladder. Somewhere your nerves are not sending the right signals either to the muscles of of the urinary system or carrying the right sensation to your brain. You have got to tell your neuroloigist about this. You likely need a referral to a urogynecologist who will likely do a testing called Urodyamics.
There is something that can add to this problem, and that is constipation. In women, especially, the stool in the rectum presses on the bladder, giving it less room. Just moving around can change the pressure within the bladder and, if the sphincter is weak can press some urine out. So people with urinary incontinence must take extra precautions to prevent or deal with constipation.
Fifty percent of MSers will deal with problem constipation during their disease.
INCONTINENCE PHYSICAL THERAPY CAN OFTEN HELP WITH THE PROBLEM
I also wrote a series of descriptions of what Incontinence Physical Therapy was like, but people haven't been very interested in reading it. Oh, well.
Incontinence PT is important to us because 50% of people with MS - women AND men - who have incontinence will benefit some from PT. The process of PT is largely to strengthen the pelvic floor. Why do we improve with PT when the problem is neurological to begin with? The answer is simply that along with the neuro deficits many of us also have lost strength in the external muscles that help us contract around the urethra, the vagina (in women) and around the rectum. We also have often developed maladaptive habits which contribute especially to urge incontinence.
After having just one episode of leaking urine at the wrong time most of us begin the pattern of emptying our bladder at the first sign of fullness. We also begin to empty our bladder more often just for convenience and before we leave the house or a comfort zone. This leads to the abnormal state of our bladders becoming accustomed to always be near empty. The result of that is that they send the signal of "full" at inappropriately levels of fullness. So we get into an increasing cycle of needing to urinate too early and having the bladder contract to remind us.
I was absolutely sure that my urge incontinence was not behavioral. Somehow, that made me feel like I was being told that my neuro problems were self-induced. But, by problems improved a great deal with therapy. They didn't completely go away, but they improved measurably.
Why go through the therapy when the problem will usually respond with drugs? The therapy might let us go longer before needing yet another drug or it might let us use a lower dose. The meds used with incontinence have many uncomfortable side effects and may actually interfere with one of the other drugs we might be on.
Hope this helps.
Quix