I'm not an English major, but I am a native speaker.
"I'll bring you home" is definitely not invalid; it's a perfectly fine thing to say, and I think your meaning is correct.
However, "I'll take you home" does not imply that you live at the same place, or that you're going to be staying over. I think it just implies a sort-of dominance on the role of the speaker. I would imagine this being said by a person speaking to someone who is more drunk than they are, or by a speaker who knows the way home better than the other person. Although, to be fair, it probably depends a lot more on who says it, how they say it, and exactly how they phrase it and not so much on bring versus take. For example, "I can take you to your place" has essentially the same meaning as "I'll bring you home."
I think the most natural thing to say in the case that you are both going back to the same place, or both heading home is "Let's go home."
There are several types of phrasal verb, and several types of verb + preposition. Not all of them are constituents, they serve different functions, they're all irregular as hell, and they're all governed by the matrix verb.
Every verb has its own assortment of special idioms, affordances, prohibitions, requirements, and irregularities. There is such immense variation in these details that such matters are considered part of the meaning of the verb; certainly they are strongly affected by the meanings. Square and cube that variation for phrasal verbs, since there are far more phrasal verbs in English than there are single-word verbs.
Some varieties can be examined in this freshman grammar homework problem. Examples of different types, from there:
Sentences (1) and (2) show two normal verb + prep constructions, from the same verb: look at, with transitivizing at; and look for, a transitive idiom meaning 'search'. Both of them require that the preposition precede the object (which may be thought of either as the object of the preposition, or as the direct object of the transitive verb + prep construction), even if that object is a pronoun. It makes no difference to most prepositions whether their object is a noun or a pronoun (ungrammatical sentences are marked with an asterisk *):
- I looked for Einstein ~ *I looked Einstein for ~ I looked for him ~ *I looked him for.
Sentence (3), on the other hand, is a real transitive phrasal verb. There are two characteristics of phrasal verbs that help to distinguish them. Both tests have limitations, however. The most important one, and the easiest test to administer, is the difference between the pattern of asterisks in the second and third columns, where pronoun objects force the difference.
There is a syntactic rule (called Particle Shift in the literature) that applies to transitive phrasal verbs only, and imposes a special requirement on pronoun objects. Thus, with a real phrasal verb like look up 'research (v)', the particle may appear either before or after a Noun object, but must appear after a Pronoun object.
- I looked up Einstein ~ I looked Einstein up ~ *I looked up him ~ I looked him up.
Note, however, that this test is helpful only with transitive phrasal verbs. There are plenty of intransitive phrasal verbs, too, but there's no object to test with. Many transitive phrasal verbs can appear also intransitively, e.g take off, move away, often with a different sense (He took it off ~ The plane took off), or not (He moved it away ~ It moved away).
The second useful characteristic is that a phrasal verb is stressed on the particle, at least as much as on the verb, and maybe more. A V + PP construction like look at, on the other hand, is stressed on the verb, not the preposition.
- He looked up the word. ~ He looked at the word.
That's because prepositions are rarely stressed, except for emphasis (In the toilet, you idiot!);
they're sposta slide by like articles and conjunctions to grease the way into the object, which is the informational part. They're not sposta distract, so they're unstressed, and therefore reduced, so we get common contractions like sposta and lookit.
Unfortunately, stress is not represented in English writing, so that distinction is not helpful for readers.
Best Answer
Apart from the fact that we would normally say '"for" the kids', rather than using 'to', both sentences are grammatically correct.
Using 'back' merely implies that you are returning to the place from which you started, and that the person speaking is at that place. If the person speaking is with you, it then becomes a matter of 'taking' something rather than 'bringing' something.
Just to clarify, let's suppose I am going to Seringapatam for the week, my wife might say:
'Don't forget to bring back something for the kids'
If, however she comes to Seringapatam with me, she might say whilst we are there
'We must take something back for the kids'.
Now if I hear that my long lost sister who lives in Seringapatam is coming to visit us, I might say to my wife:
'I hope she brings something for the kids' (Note: no use of 'back')