How do I practise grouping characters into words? At which stage of learning should one expect to be good at this?
As a native speaker, I'm trying to introspect my understanding process:
借支 is not a common term. Even in the right context (the money business), it may take a while for a native speaker to realize these two characters are meant to be a word. 笔 isn't really ambiguous as its position in the sentence dictated it has to be a noun, so 支 is the measure word and 借 is the single-character verb 'to borrow'. 借支笔 sounds right, just like the other phrases with a similar pattern that bump into my head when I read this sentence: 借张纸, 借本书, etc.
As you can see, the key knowledge here is the contextual usage and the idiomatic patterns, which is something that really takes many years to build up and comprehend. Most native speakers learn it in elementary and middle school years in both explicit (textbook) and implicit (daily conversation) ways, hardened by a lot of repetitions, homework and examinations. For example, we were taught to differentiate the two possibilities of 把手 in 3-4 grade in elementary school (at age 8-9):

把手坏了 The door handle (把手) is broken.
把手拿开 Move your hand away.

Or 球拍:

球拍 卖 完了 The racket (球拍) is sold out.
球 拍卖 完了 The auction (拍卖) for balls is over.

And our language development comes to a stage that we can say and resolve most of these ambiguous phrases smoothly, generally at age 14-15.
There is a lot of freedom in Chinese grammar and a lot of mix-and-match combinations in the vocabulary, accordingly there are a lot of idiomatic patterns to know before you can segmentate a sentence with ease.
How native speakers deal with it? If there is only one way to make sense of the sentence, other possible segmentations don't really make sense, then there is no ambiguity and nothing to avoid. If it's really ambiguous, the speaker may become aware as soon as he says it and then adds something to clarify, or the audience asks a question or just repeats the sentence in a confused tone to seek elaboration. These methods are no different from any other language.
And yes, there are things that is totally ambiguous, even native speaker couldn't tell. Like 南京市长江大桥, without context you never know which of the two it means:

南京市 长江 大桥 The Yangtze River bridge in Nanjing city
南京 市长 江大桥 The major of Nanjing - Jiang Da Qiao