How to know if a transformer will output A.C. Or DC

Although the question has provided limited details, this answer presents a somewhat different hypothesis from the standard assumption that there's an inductive coil hidden in there somewhere.

The charger in question possibly uses a Piezoelectric Transformer instead of the magnetic (inductive) transformers usually seen for isolation.

Does the charger looks somewhat like this?

If yes, the designers have used a Piezo transformer instead of a conventional one. Interestingly, the source of this image is a paper in a Korean academic publication. This makes the hypothesis even more apt.

A piezoelectric transformer designed for Mhz operation, 500 mA secondary current with 5 Volt signals, using Polyvinylidene Fluoride (PVDF) as the piezoelectric medium, could be fabricated as a thick 1210 SMD part. Since the question mentions SMD parts up to 4516 metric i.e. 1806 imperial, one of the largest of those components is probably the piezoelectric transformer providing the isolation as per the question.

Some interesting information gleaned while investigating this mystery charger:

• Piezoelectric transformers deliver 80% to (recent experimental versions) 90% efficiency, impressive in transformer terms
• These transformers can provide galvanic isolation at multi-kV levels - of course, not in a SMD 1210 size, where the contacts would be too close together.
• PVDF exhibits piezoelectricity several times greater than quartz. Hence it is ideal for making Piezo transformers.
• Many LCD display CCFL backlights are made using Piezoelectric transformers instead of the inductive coil ballast used in earlier versions. So it isn't really new technology.
• Equipment used in magnetism-sensitive areas (e.g. MRI labs) are expected to transition to non-magnetic electronics, hence Piezo transformers where a transformer is needed. (n.b. Any current flow, however, will still generate some magnetism courtesy H fields)

Some articles of interest:

• Piezoelectric Transformer With Very High Power Density
• Panasonic's Piezo transformers for LCD backlights - Of course, much bigger than the dimensions usable in USB chargers.
• A comparison of magnetic and Piezo transformers by Texas Instruments

Full disclosure: I have never worked with, or even seen Piezoelectric transformers before today - the above information was a new learning for me, in the process of investigating the mystery charger.

A transformer, as you well know, is made up of two or more coils around a core of ferrous material. That ferrous material is not a solid lump of metal, but a series of plates laminated together with adhesive. This is done because:

Early transformer developers soon realized that cores constructed from solid iron resulted in prohibitive eddy current losses, and their designs mitigated this effect with cores consisting of bundles of insulated iron wires. Later designs constructed the core by stacking layers of thin steel laminations, a principle that has remained in use.

-- Wikipedia

So you have lots of steel plates stuck together, but not only that:

Each lamination is insulated from its neighbors by a thin non-conducting layer of insulation.

Lots of metal plates, each with an induced magnetic field. That magnetic field acts between the adjacent plates stretching and squeezing the adhesive and insulation between them. Over time that adhesive starts to break apart and the laminated layers separate from each other slightly. This is the humming noise you can hear. It's always present, but once the adhesive starts to break it gets louder. These micro-fractures in the adhesive may not be visible to the naked eye, but in extreme situations they may be so bad the layers of lamination become loose and the transformer literally rattles as you shake it.

Also, the more current you draw through a transformer the larger the induced magnetic fields, and thus the louder the transformer hums (and the shorter its life span).