They are all perfectly fine. In English there is a relatively clear distinction between the following phenomena:
- plain verbs with adverbial phrases, as in “due to the rain he is going under the bridge”
- phrasal verbs as in “the boat is slowly going under”
- prefix verbs as in “the building is undergoing renovation”
Nevertheless, it is possible for a construction in one of these classes to be used so much in a specific context that it ossifies, can be used much more generally/abstractly than before, and moves on one step down on the list — often from 1 to 2, not so often from 2 to 3.
German has the same system with a slight twist:
- plain verbs with adverbial phrases, as in “ich ziehe meinen Hund unter die Wasseroberfläche / ich habe meinen Hund unter die Wasseroberfläche gezogen” (draw under …)
- separable verbs as in “ich ziehe das Strychnin in den Plätzchenteig unter / ich habe das Strychnin in den Plätzchenteig untergezogen” (fold in)
- prefix verbs as in “ich unterziehe mich einer Rosskur / ich habe mich einer Rosskur unterzogen” (subject oneself to)
The slight twist is that the non-finite forms (infinitives and participles) of German phrasal verbs are written as single words, and since this is not the case for the finite forms, these words are known as separable verbs. But just like in English, popular constructions can move downward in the list. Whenever that happens, for a while, both constructions are possible in the relevant contexts.
What you have found is contexts in which you have free choice between interpreting mit kommen / mitkommen and mit nehmen / mitnehmen as plain verbs with an elliptical adverbial phrase (the preposition mit without any referent) on one hand and as separable verbs on the other hand. I guess we can think of the first option as option 1.5 on the list — an intermediate state while the construction is moving from 1 to 2. I am sure similar observations can be made in English, though I can’t think of a good example right now.