There is no problem with installing junction boxes above a suspended ceiling, as long as the box is less than 100 in.³ and securely fastened. National Electrical Code has this to say...
National Electrical Code 2014
Chapter 3 Wiring Methods and Materials
Article 314 Outlet, Device, Pull, and Junction Boxes; Conduit Bodies; Fittings; and Handhole Enclosures
II. Installation
314.23 Supports. Enclosures within the scope of this article shall be supported in accordance with one or more of the provisions in 314.23(A) through (H).
(D) Suspended Ceilings. An enclosure mounted to structural or supporting elements of a suspended ceiling shall be not more than 1650 cm3 (100 in.3) in size and shall be securely fastened in place in accordance with either (D)(1) or (D)(2).
(1) Framing Members. An enclosure shall be fastened to the framing members by mechanical means such as bolts, screws, or rivets, or by the use of clips or other securing means identified for use with the type of ceiling framing member(s) and enclosure(s) employed. The framing members shall be adequately supported and securely fastened to each other and to the building structure.
(2) Support Wires. The installation shall comply with the provisions of 300.11(A). The enclosure shall be secured, using methods identified for the purpose, to ceiling support wire(s), including any additional support wire(s) installed for that purpose. Support wire(s) used for enclosure support shall be fastened at each end so as to be taut within the ceiling cavity.
Code Violations
In the first image, the wires being spliced together like that violates NEC 300.15.
300.15 Boxes, Conduit Bodies, or Fittings Where Required. ... Where the wiring method is conduit, tubing, Type AC cable, Type MC cable, Type MI cable, nonmetallic-sheathed cable, or other cables, a box or conduit body shall be installed at each conductor splice point, outlet point, switch point, junction point, termination point, or pull point...
I've also noted some violations in the second image.
Click for larger view
This sounds like a capacitive coupling issue -- the electric field radiated by the wires can and will couple to nearby metal objects, and a voltage-detector or high-impedance multimeter will pick it up. If you can get a multimeter with a low-impedance (LoZ) voltage range, then that can be used to discriminate between capacitively coupled "phantom" voltages and voltage sources that have some "oomph" behind them.
Best Answer
I am addressing your question first. You can't use a voltmeter to check the box by putting one of your probes to one side of the box and one on the other side. You need to kill all power to the electrical circuits in the box and using you continuity tester to "ring out" which hot wire is going to ground at the box.
That being said, replacing the metal box with a plastic does not fix the problem of a shorting circuit. It simply removes point of grounding which allows the short to trip your overcurrent device (the breaker). This means anyone who comes into contact with the box and is grounded will be subject to electrical shock or a burn hazard. So the box is actually telling you that you have a problem with a circuit inside the box and it needs to be repaired ASAP.
So your wiring is in need of repair and you should address that problem immediately. As @SDsolar has said if you are not skilled enough to make the testing and repair. You need to get a professional to do it immediately.