Unfortunatly you look to have hit the fritzing bug and corrupted the parts database for this sketch. That is likely because of the connections made in pcb view and/or some wiring errors in breadboard view. The only known solution is to start again. Here are some of the things I see wrong and suggested solutions:
1) your op amp problem is because you aren't using an op amp with sub parts. I thought the generic dual op amp had sub parts but that doesn't appear to be correct. However the ADTL082 in core parts (which you can find by entering ADTL082 in the search field and hitting enter) does and has the same pin out as the tl072. What that means is that because of the sub parts you can move one section of the op amp around independent of the other in schematic as you want to to make the schematic identical to the original.
2) You need two power supplies, but have only used one in breadboard. That also has bearing on your battery question. While the schematic calls for +/-15V in practice the op amp will take anywhere between +/-5V and +/-18V. If you want to run off batteries (as opposed to a line operated power supply) and possibly even off a line operated supply such as a wall wart one of these is a good bet:
It is possible that the device that is supplying the trigger signal may also be able to supply 5V which would fix your power problems, if not there are some other suggestions below. This device takes 5V in (you still need to get regulated 5V either from the device providing the trigger, a regulated wall wart or from a 9V battery and a 7805 type regulator) but it will give you the +/- supply voltages (at 12V rather than 15V) that you need (the original schematic indicates that either +-/15 or +-12 is fine). The above is less than $2 with free shipping (but it takes a long time to come!). Another possibility is 2 9V batteries connected in series to give you +/-9V. Transistor bias resistors may need some tweaking or may not (I'd say try it as is first and see if it works 
). On the breadboard I'd use the top two rails for +15V and gnd and the lower 2 one for ground and the other for -15v and repeat that setup on the bottom breadboard and do the interconnections as you have but with 3 wires rathar than 2 as at present. You should first do the complete schematic and then save a copy of it as an fzz file in case of later problems so you can go back to that rather than start again as is probably needed this time. Complete all the connections in schematic first and then move on to breadboard (keep a copy of the current fzz file so you can copy the layout of the current breadboard when placing the components as that seems mostly fine) and then click on a rats nest line to create the wire and drag the resulting wire in to a neat route (I like to use straight lines and 90 degree bends but whatever works for you). You probably want a second phone jack to connect to the trigger input (and I think the current phone jack is connected wrong, in schematic it is connected to the output, in breadboard it is connected to the trigger input which will cause problems in the database as it shorts the trigger input to the output). Since you only need mono you may want to use the RCA jack instead (probably the third one which has a correct mono schematic). You may want to first post you finished schematic fzz and let us look it over and see if there are any problems before trying to make the breadboard. Don't make any connections in pcb (as they will reflect back in to the other two views and cause problems if they aren't correct!)
Peter