[请你][翻译][这句话][成英文] - [Subject] + [verb] + [object] + [relative phrase] is a typical English [SVO] sentence. (please translate this sentence into English) It is not really incorrect in English grammar sense, but it is unusual in Chinese grammar.
The typical Chinese grammar is as you wrote: [请你] [把这句话] [翻译] [成英文].(Please take this sentence and translate it into English)
'把' is a 'deposit marker' that deposits the object 这句话. Thus enable the object to be placed before the verb to be emphasized.
"(把)这句话翻译成英文" = "(take) this sentence and translate it into English"
As for [不同句子的] [可以翻译成不同的词], it is a [topic] + [comment/ opinion] sentence
"不同句子的可以(被)翻译成不同的词" = "A phrase in different sentences, can be translated into different phrases"
A more detailed example:
Topic: [不同句子中的英文词] (An English phrase in different sentence (context)
Comment/ opinion: [可以被翻译成不同的中文词] (Can be translated into different Chinese phrase)
"不同句子中的英文词可以被翻译成不同的中文词" = "An English phrases in different sentences, can be translated into different Chinese phrases"
I [miss] him = 我[想念]他
I will [miss] the train = 我会[错过]火车
Which are other examples of situations/ structures where you can use 翻译 without 把? Is there a rule?
You can use 翻译 in a SVO sentence without 把
Both sentences below are correct:
把这句话翻译成為英文並不容易 (take this sentence and translate it into English is not easy) -- use 'deposit marker' 把 to place the object '这句话' before the verb to be emphasized.
翻译这句话成為英文並不容易 (translate this sentence into English is not easy) -- using straight SVO structure without emphasizing the object