May one use a toilet that contains toilet soap?
One is to avoid using all forms of toilet soaps on Shabbos. Thus one is to remove the soap from the toilet before Shabbos. If one did not do so it is permitted to remove the soap on Shabbos. Some are stringent to remove it with an irregularity. Some Poskim however rule it is permitted to use such soaps in cases of discomfort. Furthermore some Poskim rule it is only forbidden to use toilet soaps if one intends that the soap color the water or that it release a good smell. If however one does not have intent for this to occur one may use toilet soaps even if they color the water and release good smell.
Urinating on the soap: In all cases it is permitted to urinate in a toilet that contains toilet soap so long as one does not urinate directly onto the soap.
Is the soap Muktzah?  Toilet soap is considered Muktzah Machmas Issur. Some Poskim however consider it Muktzah Machmas Chisaron Kis. According to this latter opinion one may not enter or remove the toilet soap from the toilet on Shabbos due to the Muktzah prohibition. It is however permitted to move it with an irregularity.
May one clean a dirty toilet?
Yes. One may even use the toilet brush to do so.
 Orach Yisrael 35
Background of the prohibition:
The use of toilet soaps may involve any of the following three prohibitions: Molid Reiach and Nolad and Tzoveia [dying]
- Prohibition of Nolad: Seemingly it is forbidden to use toilet soaps due to the Nolad prohibition being that one is melting the solid soap thru flushing the toilet [which is similar to urinating on snow which was viewed as if one is melting the snow with ones hands]. Thus even using unscented and non-colored toilet soaps would be problematic. [Orach Yisrael 35 rules based on this reason that initially it is forbidden o use toilet soaps unless it is a case of Shaas Hadchak] However there are Poskim which rule that there is no issue of Nolad with toilet soaps as it is only a minute amount which melts, and that itself is through an indirect action.[Shabbos Kehalacha Vol. 3 17/80]
- Prohibition of Tzoveia: If the toilet soap releases a color into the water, then it may not be used due to the dying prohibition. [SSH”K 23/14] However some Poskim allow its use if one has no interest at all in the coloring of the water. [Shabbos Kehalacha Vol. 3 17/80]
- Prohibition of creating new smell: The prohibition of Molid Reiach does not apply even when using a scented toilet soap. [Practically although Poskim [Sharreiy Teshuvah and Mishneh Berurah in 128/6.Minchas Yitzchak 6/26] rule it is forbidden to create a good smell in water, nevertheless it is allowed to absorb a good smell into the toilet water as the good smell is simply there to remove bad odor. [Beir Moshe 1/34 rules that good smells may always be applied in order to remove bad odors, and so rules Piskeiy Teshuvos [327/1, 328/26] Other Poskim however rule it is only allowed if one has no intent to create a good smell within the water. [Shabbos Kehalacha Vol. 3 17/80]
 See below and Shabbos Kehalacha Vol. 3 17 footnote 196 for a discussion on the Muktzah status of the soap.
 Orach Yisrael 35; Shabbos Kehalacha ibid, there he extends the leniency to even toilet soaps that dye the water.
In Sefer Orach Yisrael 35 he rules that one may be lenient to use such soaps if lack of doing so will cause bad odors and distress, as he rules that in a Shaas Hadchak one may be lenient like the opinion which holds that melting soaps and fats do not contain a Nolad prohibition. Furthermore he rules that if one forgot to remove it before Shabbos then it may be used on Shabbos being that it is Muktzah and is forbidden to remove and thus this case too is a Shaas Hadchak. Vetzaruch Iyun according to Admur why he rules by melting fats near fire that one is to be stringent unless it is a Shaas Hadchak while here by soaps he plainly rules that one is to be stringent [implying even in Shaas Hadchak].
 Shabbos Kehalacha ibid
 Shabbos Kehalacha Vol. 3 17/81
 Shabbos Kehalacha Vol. 3 17 footnote 196
 Igros Moshe 5
 SSH”K 23/15