"Raj takes food from Neha."
Translation:राज नेहा से खाना लेता है।
This is similar to the other sentence you asked about.
You'll hear this kind of phrasing in colloquial Hindi all the time but in formal Hindi, the direct object must be placed closer to the verb.
The direct object is basically the noun phrase that answers the question क्या [verb]?. For example, in this sentence, if you ask क्या लेता है?, the answer is खाना so that's the direct object.
Again, like in the other sentence, you can say राज खाने को नेहा से लेता है। by adding a को to the direct object but this phrasing places a lot of emphasis on the खाना.
In the verb infinitive+है construction like in 'राज को नेहा से मिलना है', the subject must be in the dative case. For nouns, this just means that they must be accompanied by the postposition को. The sentence would literally translate to something like 'It is required of Raj to meet Neha' or 'It is wanted by Raj to meet Neha'.
'राज नेहा से खाना लेता है' is the one-one translation of 'Raj takes food from Neha' so the subject is in the direct case.