為(ため)にFor, For the sake of, In order to

These share the same core construction, using ため(に) to express 'in order to (A)' or 'due to (A)'. ため(に) is used within a sentence after a noun, verb, or adjective phrase. そのため is used at the beginning of a sentence to mean 'for that reason' or 'due to that', referring back to what was just said.

Both can mean 'in order to' when expressing purpose and are often interchangeable in that usage, e.g. 健康維持のために運動する / 健康を維持するのに運動する ('I exercise in order to maintain my health.') However, ために can also express cause in certain contexts, which のに cannot, e.g. 雨のために試合が中止になった ('The game was canceled due to rain.')

わけだ and ため(に) are similar in that both can be used when expressing a reason, but they mean quite different things. わけだ is used when drawing a conclusion, like the English 'so that's why (A)', while ため(に) on the other hand means 'for the sake of (A)' or 'due to (A)'.

Both can be translated as 'for (A)', but differ in nuance. 向き attaches directly to a noun and suggests suitability, e.g. 子ども向きの本 ('A book suitable for children') ため(に) connects to nouns with の and suggests intended benefit or purpose, e.g. 子どものための本 ('A book for children [designed to benefit them]')

んがため(に) can be considered an old-fashioned, literary form of ため(に). While both can express the idea of 'for the purpose of (A)' or 'in order to (A)', for example 生きんがために食べる 'eat (in order) to live' sounds like a much more formal and poetic version of 生きるために食べる (same meaning).

Both can express purpose, similar to 'in order to' or 'for that reason', e.g. 海外に行くためにパスポートを取った ('I got a passport in order to go abroad.') 海外に行くにはパスポートが必要だ ('A passport is necessary to go abroad.') Unlike ため(に), には also has broader uses, such as marking scope or perspective, e.g. 子どもには難しい ('It's difficult for children.')