No, Islam is not banned in Japan. The wikipedia entry about Islam in Japan says:
The history of Islam in Japan is relatively brief in relation to the
religion's longstanding presence in other nearby countries. Islam is
one of the smallest minority faiths in Japan, having more adherents in
the country than the Bahá'í faith, but fewer than Christianity.
Japan's Muslim population is around 70,000 which includes less than
10,000 of them as Japanese and the rest being foreigners (Majority of
them are nationals from Southeast Asian countries).
The constitution of Japan states in its article 20:
Freedom of religion is guaranteed to all. No religious
organization shall receive any privileges from the State, nor exercise
any political authority.
Reference: Ministry of Justice, Japan.
According to Japan Times, there are more than 100,000 Muslims and 60 mosques in Japan.
An Egyptian legal scholar based in Ibaraki, Osaka Prefecture, says he
is working to increase understanding between the Japanese people and
the more than 100,000 Muslims who live among them.
Islam has gradually put down roots in Japan. There are now around 60
mosques, many of them established in former private homes.
Reference: Japan Times
And there are no restriction on preaching Islam in Japan. Recently, Dr. Zakir Naik, a Muslim preacher, held a big conference where some people converted to Islam. One of the conferences (the topic was "What is Islam") were held in Kyushu University which is owned by the Japanese government. The authorities gave him permission to hold the conference and didn't interrupt the event for violating any laws.
So, Islam is not banned in Japan, Promotion of Islam is not banned in Japan, switching into Islam is not banned in Japan and the Japanese government is in fact, quite friendly to Islam (compared to western countries).
Edit: The OP asked some sub questions that I didn't address. I will try to address them all in this edit.
Can someone confirm or disprove if Muslims indeed cannot get permanent residency and citizenship?
Yes, Muslims can get Japanese citizenship. Japan has citizens who came from predominantly Muslim countries. And Japan doesn't list the religion of its citizens.
According to 2008 Japanese government figures, 19.9% of registered
Pakistanis lived in Saitama, 17.8% in Tokyo, 12.3% in Kanagawa, 10.4%
in Aichi, 8.98% in Chiba, 7.59% in Gunma, 6.02% in Ibaraki, 4.44% in
Tochigi, 4.21% in Toyama, 3.27% in Shizuoka, and the remaining 4.98%
in other prefectures of Japan. Only an estimated 200 Pakistanis
hold Japanese citizenship.
Is it true that there is a strong ban on the propagation of Islam in Japan?
Many Japanese wives of Pakistani migrants have converted to Islam, and in fact form the largest group of native Japanese converts to Islam.
Are Arabic languages taught at Japanese universities?
Yes. Andrew Grimm's answer has more on this.
Is halal slaughter legal in Japan?
Yes. In fact, local authorities of a Japanese town named Taito Ward subsidies halal food.
In October 2015, Taito Ward began to provide up to ¥100,000 in
subsidies to local businesses who wished to obtain halal
certification. According to Takuji Kawai, director of the ward’s
Tourism Section, applications were filed so quickly that the ward had
reached its quota in just two months, and were forced to double it.
What started out as three halal-certified restaurants and shops in
Taito Ward, including Naritaya, became 19 in just a few months. In
December, the ward also printed 10,000 copies of a Ueno/Asakusa map
for Muslim tourists that show mosques, halal-certified restaurants,
and Muslim-friendly hotels and souvenir shops in the area. The map
went out of stock so fast that an additional 10,000 copies have
already been ordered.
Reference: Japan Times