Why are there so many terms for these generic words?
Most of them have defferent meanings. Some are used for Western royalty, Some are used for Chinese royalty.
王 - (in general) King e.g. アーサー王 King Arthur
王 - (in historical China and historical Asia) One of the titles of the loads in the Imperial China, or the King of the Imperial Chinese tributary state
国王 - (as in the Head of State, especially in present‐day) King e.g. フアン・カルロス1世国王 King Juan Carlos I, タイの国王ラーマ9世 Rama IX the King of Thailand
王様 - (colloquial, reverential, from people's point of view) King e.g. 王様のお城 the King's castle
君主 - Monarch (This is not a honorific.) e.g. 絶対君主制 absolute monarchy
大王 - King xxx the Great e.g. アレクサンダー大王 Alexander the Great
大君 - (in historical Japan) Diplomatic Title of Shogun (将軍 is a General, but de facto King in Japan. However 王 meant a vassal of China in those days, so they coined this Title.)
帝[みかど] - (in historical Japan) Emperor of Japan
帝[てい] - Honorific of the emperor (except for Emperor of Japan)
皇帝 - (except for Emperor of Japan) Emperor e.g. 中国の皇帝 the Emperror of China, ローマ帝国初代皇帝 first Emperor of the Roman Empire
帝王 - Crowned heads, Shah of the Old Persia, (idiomatic) King e.g. ジャズの帝王 King of jazz
大帝 - Emperor xxx the Great e.g. オットー大帝 Otto the Great
天子 - (in historical China) Emperor of China, (in historical Japan, before 10th century?) Emperor of Japan
天皇 - Emperor of Japan
天王 - (in historical Japan) alternative spelling for 天皇
女王 - Queen (female King) e.g. エリザベス2世女王 Queen Elizabeth II
王妃 - Queen (King's wife) e.g. スウェーデンのシルビア王妃 Queen Silvia of Sweden
皇后 - Empress consort (Emperor's wife)
皇太后 - Empress (former Emperor's wife, Emperor's mother)
太皇太后 - Empress (former Emperor's wife, Emperor's grandmother)
妃/后[きさき] - (informal, outdated, except for Japanese) Imperial wife
お妃様/お后様 - (colloquial, reverential, from people's point of view) Imperial wife
女帝 - Empress regnant (female Emperor)
王子 - (in general) Prince (Children of royalty) e.g. ウィリアム王子 Prince William
皇子 - Prince (Emperor's son)
皇太子 - Crown prince e.g. 日本の皇太子 Japanese Crown prince, チャールズ皇太子 Prince Charles
親王[しんのう] - Honorific of the Japanese Emperor's son, Honorific of the Japanese Emperor's brother
王女 - Princess (King's daughter) e.g. アン王女 Anne, Princess Royal
皇女 - Princess (Emperor's daughter)
妃 [ひ] - Honorific of the Imperial wife e.g. ダイアナ妃 Princess Diana, キャサリン妃 Princess Catherine
親王妃 - Princess (親王's wife)
内親王 - Honorific of the Japanese Emperor's daughter, Honorific of the Japanese Emperor's sister
王子妃 - Princess (王子's wife)
皇太子妃 - Princess (皇太子's wife) e.g. 皇太子妃ダイアナ Princess (wife of the crown prince) Diana
妃殿下 - 妃 + 殿下 (Her Royal Highness)
大君女 - (in historical Japan) Princess (daughter of the Japanese Emperor)
姫 - (informal) Princess, Noble's daughter, shogun's daughter, daimyo's daughter
お姫様 - (colloquial, reverential, from people's point of view) 姫
姫御子 [ひめみこ] - (in historical Japan, before 10th century?) Princess (daughter of the Japanese Emperor)
If my assumptions from above are correct, which ones are for Japanese royalty and which ones are for foreign royalty?
天皇 is exclusively used for Japanese Emperor. 皇后, 皇太子, 皇太子妃, 皇太后, 太皇太后, 皇子, 皇女, 親王, 内親王 are used for Japanese royalty (not exclusively, though). 国王, 女王, 王妃, 皇太子, 皇太子妃, 王子, 王女, are used for foreign Kingdoms.
which ones are religious terms and which are "regular"?
There are no religious terms. (However,The Emperor of Japan is closely related to the Shinto religion). 大君, 帝, 天子, 天王, 大君女, 姫御子 are obsolete.
What are the rules for being able to apply 陛下 or 殿下?
You can use 陛下 with King, Queen, Emperor and Empress (国王陛下、女王陛下、王妃陛下、天皇陛下、皇帝陛下、皇后陛下). Other royalty are 殿下.