I don't want to whine too much about safety and I don't like kilometers long safety notices which don't say anything, but I feel that my behavior is justified in this case, so here it goes: If you're uncomfortable with opening the supply, DON'T USE IT AT ALL!!! IT COULD BE VERY DANGEROUS!!! I'll try to explain a bit later why.
I'm actually planing to modify a PC power supply for my project too, so I have several useful links to share.
Take a look at following links:
http://web2.murraystate.edu/andy.batts/ps/powersupply.htm
http://www.wikihow.com/Convert-a-Computer-ATX-Power-Supply-to-a-Lab-Power-Supply
http://www.instructables.com/id/Convert-an-ATX-Power-Supply-Into-a-Regular-DC-Powe/
http://www.instructables.com/id/Yet-Another-ATX--Lab-Bench-Power-Supply-Conversion/
http://www.mbeckler.org/powersupply/
http://www.marcee.org/Articles/PCPowerSupply.htm
They show several projects where people used ATX power supplies for home experimenting.
I'd also recommend that you read some basic theory about switching power supplies. It will be clear to you then why yours isn't starting. I don't have any interesting links at the moment, but here's a start.
Basically, for SMPS to work, they need a minimum load. That's why yours isn't starting. When power supply is correctly connected to a computer, the computer will provide minimum load to start the supply. As you can see, if you follow the links above, people get around that by providing a low resistance resistor with high power rating and connecting it to the supply. The resistor will provide load for the supply to start. Some articles say that one on the +5 V line is needed, but on some supplies a resistor may be needed on each of the power lines. That's one of the reasons why you should open the supply and take a look how it's working inside. You should be able to identify which cable is connected to which power line. If you can't then I'd recommended that you do some more researching before doing anything. Some articles mention that they use 10 Ω 2 W resistors on + 5 V. I'd use a bit stronger resistor, so its power rating wouldn't be exceeded (but articles also mention that resistors are cool).
About sensing wires: They are there to allow supply to regulate itself and some supplies may have as much as one for each power line. Articles I linked talk about them and on how to identify them. If I'm correct, they should already be connected to the 24 pin connector and ready to work, but as you mentioned, you didn't connect anything to the connector, so it's not surprising that they are making problems. You'd have to modify PSU cable if you want to solve the problem with them, but if you're still uncomfortable with opening the supply, I'd recommend that you don't try modifying the cable.
As for detecting if the supply is working correctly, use the PWR_OK pin and a voltmeter. You should get +5 V once the supply has stabilized itself.
It should be possible and not too complicated. There are many guides explaining the process. Here's one.
UPDATE:
To complement the other answer:
Some computer PSUs may already have needed load installed and can provide stable voltages out of the box, but you can't rely on that.
The instructable recommends use of $10\mbox{ }\Omega\mbox{, 10 W}$ resistors for $\mbox{+5 V, +12 V}$ and $\mbox{-12 V}$. The $\mbox{12 V}$ resistors are going to be dissipating around $\mbox{14,4 W}$, so my recommendation is to get some bigger resistors, just in case.
The other option would be to use resistors with higher resistance for the 12 V lines, for example $22\mbox{ }\Omega$. They'll use less current, but I think that it won't be a major problem because the screen will use considerable amount of power too.
This isn't mentioned in the instructable, but I'd put some $10\mbox{ }\Omega$ or $22\mbox{ }\Omega$ resistors on the $\mbox{+3.3 V}$ line too.
Another part that has been omitted is how to locate the power supply rails inside the power supply itself.
Here you can find the color code for computer supply wires. Just follow the wires inside the case of the supply and see where they end. The wires you're interested in are yellow for $\mbox{+12 V}$, orange for $\mbox{+3.3 V}$ and red for $\mbox{+5 V}$.
Don't forget the LEDs for gray and purple wires. You can use tool like http://ledcalc.com/ to get the correct resistor for the LED.
Best Answer
You can try it, sure. Your pictured power supply appears to have common returns so an ATX power supply may work, so long as the rise times and sequencing of outputs are OK.
If the ATX power supply is much stronger than the one you're replacing (higher current) and you have some abnormal condition like an overload or a short, the stiffer power supply may end up further damaging the load.
Many safety-approved items say "use only model X power supply" on their labels. Be careful if you see something like this.
If your end equipment does work with the ATX power supply, you at least know where the fault lies (i.e. the power supply) and may wish to either source an identical replacement or attempt a repair.