When looking up the word "with", I found the following translations:
qum. But I am unsure of what the particular semantic and syntactic differences are for these words. Where would they be used and what are their differences?
With is such a versatile word in English that how's it actually being used is sometimes obscured. Consider the following:
I am going to the store with my friends.
I am making a house with the best tools.
With is used in these sentences, but they're using it very differently. The first expresses accompaniment: the friend are coming along. The second expresses means: the job is accomplished by means of an object.
There is no one Latin word that encompasses all the varied uses of with. There is, instead, a combination of uses. This is typically accomplished by three different constructions.
Note: some of the examples I give come from Moreland & Fleischer or Gildersleeve & Lodge.
The first is the ablative of means, as mentioned above. For expressing "with = by means of", all you need is a simple ablative.
Nautae gladiis pugnant. | The sailors fight with (= by means of) swords.
The second is the ablative of manner. Like the ablative of means, the noun within the prepositional phrase is an ablative, but this time it often has the preposition cum to go with it. Ablatives of manner express the way or, as you guessed it, manner in which something is done. Essentially, if you can replace it with an adverb, it's likely an ablative of manner.
Verba misera cum venia audivisti. | You listened to my wretched words with indulgence.
Note that you can substitute "with indulgence" or "indulgently", but you cannot do that for the means or accompaniment "with."
Note: if you further modify the ablative of manner with an adjective, it can drop the cum:
Verba misera magna (cum) venia audivisti. | You listened to my wretched words with great indulgence.
The third way to express with is with the ablative of accompaniment. This is the first one I explained above. It also looks identical to ablative of manner, except that you cannot replace it with an adverb.
Domum cum amicis eamus. | Let us go home with our friends.
In English, sometimes qualities of a noun are denoted with with:
There's that girl with the ugly nose.
In Latin this is typically constructed with the ablative, though sometimes the genitive is used instead:
Est ista puella naso turpiculo.
Gildersleeve notes about the distinction between the ablative and genitive here:
External and transient qualities are put by preference in the Ablative ; Measure, Number, Time, and Space are put in the Genitive only ; parts of the body in the Ablative only. Otherwise there is often no difference.
While there are other uses of with in English, this should give you a good sense of what is common.
One last note, though, about the words you listed.
First, qum is a misspelling of archaic forms of cum, both quom and quum being used at different times (even as late as the first century BCE).
Apud can sometimes substitute for the ablative of accompaniment—apud me = "with me"—but typically it refers to location. Therefore it answers ubi (where?), whereas the ablative of accompaniment answers *cum quibus" (with whom?). This way, it also can refer more generally to things associated with the noun it precedes. Apud me can mean "with me," "at my house," or even "in my book" (though typically the latter is only used when referring to someone else, so apud Herodotum = "in the works of Herodotus").
Per plus the accusative can substitute for the ablative of means, but the means are less physical and more abstract, and usually impute some sort of authority or intentionality. You can typically use "through" for per, though.