1)
It is prohibited to launder clothing, or even to wear freshly laundered
clothing or use freshly laundered linen, tablecloths and towels, during
the nine days, except for on Shabbos. (Shulchan Aruch and Rama Siman 551:3)
It
is similarly prohibited to give clothing to a non-Jewish dry cleaner or
Laundromat (or a non-Jewish housekeeper) in this time period, even if
the clothing isn't needed for the nine days, and will only be picked up
and worn after Tisha B'Av. Clothing that is needed for very small
children, may be laundered in small loads, as needed. (Rama ibid. and Mishna Berura S"K 83)
If
a Jew owns a dry cleaners or a Laundromat, he may clean clothing of
non-Jews during the nine days, if his Parnassah depends on it. (See Mishna Berura Siman 551 S"K 42)
2)
It is the accepted custom to prohibit showering the entire body at once
in the regular manner during the nine days, even with cold water,
besides for a shower on Erev Shabbos. (Rama Siman 551:16 and Mishna Berura S"K 94)
If
one is unable to go nine days without a shower, as is the case for most
people nowadays, certainly in the United States where most people
shower daily, there are various leniencies discussed by the Poskim. Some
allow showering without soap & shampoo. Some Poskim allow only cold
showers (or at least not as hot as one is accustomed to).
Obviously,
for medical reasons regular hot showers are permitted. Thus, pregnant
women, mothers of newborns, and other frail people may shower as
necessary. (See Sha'ar HaTziyun Siman 551 os 94. According to
many Poskim, one who is extremely sweaty may also shower, with cold
water, to freshen up without shampoo and soap, as this isn't considered
washing for pleasure. See Aruch Hashulchan Siman 551:38 and Igros Moshe
Even HaEzer Vol. 4 Siman 84:4)
One
who has the custom to immerse in a Mikvah every day, without fail, may
immerse in a Mikvah during the 'nine days' provided the water is not
hot. (See Aruch HaShulchan Siman 551:35)
Likewise,
one who has the custom to immerse in a Mikvah each Erev Shabbos,
without fail, may do so on Erev Shabbos Chazon as well, provided the
water is not hot. (Mishna Berura Siman 551 S"K 95 and Sha'ar HaTziyun os 98)
For Halacha L'Ma'aseh, as always, a Rav must be consulted.