というか is a word used before trying to rephrase/re-explain something. It's close to "or rather" or "I mean" in English.
～という意味では literally means "in the sense of ～", but here you can roughly translate this as "as far as ～ goes", "if you mean ～", or "speaking of ～", etc.
So he first tried to say 最近の日本のプロレスのスタイルという意味では ("regarding the recent style of Japanese pro wrestling"), but stopped saying it in the middle and rephrased it to something more specific, 危険な技の応酬という意味では.
Speaking of the recent style of Japanese pro wrestling... I mean, (speaking of) the exchange of dangerous moves in particular, (yes, certainly,) there have been such a trend.
And this sentence:
Each wrestler must (re)think about (things like) dangerous moves and reckless games these days. (lit. "games about which you don't consider risks")
だったり/であったり is just another way to list nouns and na/no-adjectives non-exhaustively. It's basically the famous
verb + たり + verb (+ たり) construction, except that the copula だ is used instead of normal verbs. 各レスラー is the subject of 考えないといけない. 今の危険な技 is a noun phrase meaning "dangerous moves (found) these days." If you were disturbed by the comma after だったり, please keep in mind that Japanese commas have very little grammatical role.