How does one understand the [verb] + 在了 grammar structure? From random Internet sites, we can see many examples of the grammar structures:

[verb] + 在了


手机忘在了出租车上。
我站在了舞台上。
因为人不在，快递员就放在了门口。
小男孩被父亲故意丢在了超市。
闺蜜和男友睡在了一起。
(Google the quotes surrounded by quotation marks to see the source(s).)

I think I get the gist from these examples.  It looks like adding 在 modifies a verb, e.g., 忘 = &quot;to forget&quot; turns into 忘在 = &quot;to forget at&quot;.  The 了 is the completion 了, added after the modified verb 忘在了 (so we don't write 忘了在 [or do we?]).  在了 doesn't seem to be used in the same way without a preceding verb (but we can say 不在了 or 没在了 to mean &quot;not here&quot;, sometimes as a euphemism for being dead).
However, there may be a better way to understand this.  I'm wondering if there is a more rigorous treatment available.  I didn't find a reference from Google, not at the Chinese Grammar Wiki.
Question: How does one understand the [verb] + 在了 grammar structure?
在了 is very common, even colloquially, but it'd be better to see it as separate 在 and 了.
The usage of 了 and its position in a sentence is complex. In the example sentence

手机忘在了出租车上,


忘 here can be understood as a labile verb (perhaps?), meaning &quot;be forgotten&quot;.
在 is a 结果补语 (complement of result).  Accroding to 刘月华(Liu Yuehua) et al.'s 《实用现代汉语语法》(Practical Modern Chinese Grammar, I'll refer to it as Gr from now on), the function of 在 is to &quot;表示通过动作使人或事物处于某个处所，后边一定要有处所宾语&quot; (convey the idea that something or someone is placed in some position via an action, and must be followed by an object of location).
出租车上 is the 处所宾语 (object of location) of 在. The object 出租车, or taxi, usually can't convey the idea of location on its own, but can do so when compounded with the adverb of position 上.
了 is a 动态助词 (aspect particle).  According to Gr, it's a bit complex; for a verb like 忘 in the sense of &quot;to leave something at some place&quot;, 了 means the action has happened and has ended.

I think your problem here can boil down to where should the article 了 be in this sentence?
According to Gr, &quot;如果谓语动词后有结果补语，“了”位于结果补语后：动词＋结果补语＋“了”＋宾语&quot; (if there is a complement of result after the verb, then 了 is placed after the complement of result: verb + complement of result + 了 particle + object). So,
手机(Subject)忘(Verb)在(Complement)了(Particle)出租车上(Object)。
This rule can explain the first four sentences you provided. I think the fifth one is the same, but I'm unsure for now.
Also, please note that in every sentence you provided, 了 can be placed at the very end. This kind of 了 is called a 语气助词 (Modal particle), and the meanings of some sentences will change when you move 了 to the end, and some will not change very much. However, this is a much more complex grammar mischief of Chinese and I don't consider myself capable of talking about it.
Hope this helps.