It appears to want (but doesn’t show the source of) +15V, and -15V both referenced to ground. For your real circuit a couple of regulated 15V wall warts (one connected normally, the -15v one with the +15v wire connected to ground and the ground connection of the wall wart connected as -15v). There are other options available as well (such as a single 48V non regulated wall wart and 7815 and 7915 regulators to get the needed ±15 supplies) if you tell us what is important, such as size, cost, only one connection to the power line (the suggested solution needs 2 sockets which may not be acceptable) or something else. You are correct that the +15V supply will go to one power rail, ground to another power rail and -15V to a third power rail or you can just jumper the various pins that need power to each other (although using the busses is usually easier) any of those should work.
Correct a connection is a connection either via the ground bus or by direct wire between pins. Note that the rats nest lines take the shortest (not always best) direct route between pins.
As noted above the rats nest line is the shortest route between pins, it ignores things like resistors. What may be easiest is to click on the rats nest line then drag the resulting wire away from the direct route. That should show you exactly where the end points are. One potential difficulty is that if you click on a part of the rats nest that is over a connector you may accidentally make an unintended connection, so its best to check the schematic to make sure it agrees there should be a connection there. As well you are welcome to post your sketch .fzz file via the upload butting (7th from the left on the reply tool bar) and we can have a look at your sketch and provide advise and/or corrections. In general a rats nest line that appears to short a resistor is probably either an error or just passing over the resistor to a different connection (which creating and moving the wire should show).
Each ratsnest line is a different net which is the collection of all pins that should be connected together, so yes that will turn in to a trace in PCB view to connect all the pins together.
It is likely more the graphic program finding a “closer” (for its idea of close) route to get between the two connected points.
What size? 1/4 inch or one of the 3.5 mm types. There doesn’t look to be one in core parts but I think there is one for at least the 1/4" type jacks available on the net (someone else is working on a midi project and needed 1/4" mono). If you get the size, a google search with a search term like “fritzing part mono jack” may turn up something. One last note about a Fritzing bug: it is best to make all your connections in one view (sounds like schematic would be it in this case) and then use the rats nest lines to route the other two views. If you make changes in more than one view the parts database sometimes gets corrupted and you get rats nest lines that shouldn’t be there. The only solution to that is to start again which can be painful (which makes it a good idea to keep a copy of the project when it is complete in one view so you can at least restart there if you have to).