Which one of the words above is most suitable in the following sentence and why?
The correct answer is 楽しげに. This げ roughly means "-looking", and it attaches mainly to a relatively small set of adjectives that describe emotion. 楽しげ is listed as a standalone na-adjective in jisho.org. Similar words include 悲しげ, 眠たげ, 嬉しげ and さびしげ. See also: What is the usage of 〜げ and how does it differ from 〜そう or 〜っぽい? and Difference between 〜げ and 〜そう
がち and 気味 are wrong because they only attach to a noun or the masu-stem of a verb.
-っぽい can attach to many words including i-adjectives:
When it attaches to the stem of an i-adjective, it forms another i-adjective with an added nuance of "-ish" or "-looking". It's not used with adjectives of emotion. っぽい used like this is not a productive suffix, and you cannot combine an arbitrary i-adjective with っぽい. We can say 安っぽい and 荒っぽい, but not 高っぽい nor 優しっぽい.
(As an exception, you can use っぽい after the dictionary-form of arbitrary adjectives and verbs, although this sounds fairly colloquial: 合ってるっぽい "looks correct", 高いっぽい "looks expensive", 泣いてるっぽい "seems to be weeping". See: 安いらしい, 安いそうだ, and 高っぽい are incorrect?)