Are anime and manga subsidized or actively used by Japanese government to promote Japanese culture/relations internationally?
The Japan National Tourism Organization (JNTO) has sponsored many programs that uses anime to promote culture and tourism. Last year, they published an English-language "Japan Anime Map" that highlights the locales in the country with ties to anime.

The map provides info on the various real-life settings in animes such as Lucky Star, The Melancholy of Haruhi Suzumiya, true tears, and Summer Wars.

It also lists anime-related museums and theme parks such as Sanrio Puroland, Studio Ghibli Museum, and the Kyoto International Manga Museum.

Info on anime-related shopping areas such as Tokyo's Akihabara, Nagoya's Osu Electric Town, and Osaka's Nipponbashi (aka Den Den Town). Also included are shopping suggestions for souvenirs to bring back such as Gundam plastic models, soft vinyl toys, and Nendoroid figures.

Naturally it also provides insight on on otaku culture topics such as cosplay, figures, and games, and also anime-related events such as Comic Market and World Cosplay Summit

The JNTO had previously collaborated with the Hakone Tourist Association to produce the "Evangelion Hakone Instrumentality Map: English Version," and the "Cool Japan Poster: Hakone" to cover the town of Hakone near Tokyo.

One of the more noticeable ones is the Young Animator Training Project.

In 2010, the Agency of Cultural Affairs of Japan invested 214 million yen (over 2 million USD) in the so-called "Young Animator Training Project" and entrusted the execution of the project to the Japan Animation Creators Association (JaniCA). 

In the following years JaniCA produced several original anime in cooperation with different production studios, continuing to receive money from the Japanese government. Each of the single episode anime was 23 minutes long. Young animators receive on-the-job-trainings under the supervision of professional anime creators as the works are shown on TV and/or at anime events. 

One of the reasons that the Agency for Cultural Affairs is supporting this initiative is the concern that more of the Japanese animation process is being outsourced overseas -- thus leading to a decline in opportunities to teach animation techniques within Japan.

For those interested, here is the trailer for the four shorts for Anime Mirai 2013 Young Animator Training Project, scheduled to premiere March 2.