Has there been research done about which languages are generally the easiest to learn for an average native Japanese speaker? Do they have an easier time learning Mandarin than learning a Western language like English?
This language school's website has hour estimates for how long it takes a Japanese speaker to learn various other languages based on their internal studies. I've attempted to translate it.
The course names are somewhat confusing but they basically boils down to something like
learn the minimum level needed when working/living abroad,
learn to be able to do basic daily communications,
learn to use daily expressions and be able to live smoothly abroad.
Note also the times are in hours of (classroom) instruction.
Course Name 'Working Abroad Course' 'Training for the Fundamentals Course' 'Course for Studying Abroad'
Spanish 50～60 120～180 240～300
French 60～70 130～150 300～350
German 60～70 130～150 300～350
Russian 100～120 180～220 350～400
Indonesian 40～50 100～120 180～240
Korean 40～50 100～120 180～240
Chinese 50～60 120～160 240～300
Thai 80～100 140～180 280～340
Note that English isn't in the table, but it is included in a difficulty chart on that website as being in the same category as French, German, Hungarian, and Thai. So, estimating based on the other languages, English probably would take around 300+ hours of instruction to reach the maximum level taught at this language school.
Japanese is considered a "language isolate" which is a language that has no demonstrable relationship with other languages or is descendant from any other language. Other examples of this is Georgian, Korean, and Etruscan.
There have been attempts to show that Japanese is related to such languages as Ryukyuan, Tibeto-Burman, and Mon-Khmer but inconclusively. Korean has been shown to be the strongest possibility but it remains a hypothesis. Strangely, I know several Koreans who can speak Japanese but this may be due to the close proximity of the two countries and the intermingling due to travel.
This leads to the conclusion that if Japanese is an isolated language it would be difficult for a Japanese native speaker to learn any other language but not impossible
Reference: Martine Robbeets (2005): Is Japanese Related to Korean, Tungusic, Mongolic and Turkic?. (Dissertation; see the abstract on linguistlist.)
While the results of this paper aren't explicit in their conclusion with respect to your question, you can infer that if an LFG can be ported from Japanese to Korean with high success, then there must be a high degree of similarity between the two languages (for some definition of high).
This is commensurate with the fact that there's a large amount of historical influence of both China and Japan within the Korean peninsula.
When hangul was invented by King Sejong in the 15th century, there was still a desire for some degree of cultural unification with China (hence influencing the maintenance of hanja, or Chinese characters, in Korean); however, Japanese occupancy in Korea also had a hand in the melding of Korean. In fact, in pre-WW2 Korea, Japanese was a mandatory course to be studied by students leaving Korean as an optional course.