As far as the motor oil leak; The oil pan is bolted to the bottom of the engine block - just like on 99% of other cars. There are perhaps a dozen bolts up through the sides of the pan into the bottom surface of the block; plus a couple more (I think) that bolt sideways into the lower part of the transmission. Getting the pan off is sort of a dirty job - but not rocket science. You would want to get the front of the car elevated -safely-. This could be with a pair of the ramps sold in auto parts places, or; I have jacked up our Neon, using the jack that came with it, and stacking wooden and cement blocks under the side of the car (right below the front door hinge area). I work one side at a time, and use good wheel chocks in front of and behind the rear wheels, to keep it from rolling away. Since once a front wheel is off the ground, being in 'park' doesnt help: The car will easily roll down any slight slope. The wooden pieces I use are 6" x 6" and 4" x 4" chunks about 2 feet long. I dont use jackstands, since my driveway is gravel, and kind of soft. A jackstand could sink in and drop the car on me.
Its not too likely the oil pan has a hole in it, though of course thats not impossibe. More likely leaks are because: 1. The drain plug is bad, with stripped threads or cross threaded. You can get new drain plugs, and should also use a new drain plug 'washer', made of plastic. 2. There is an oil sealing gasket between the oil pan and the block. This can get really soft with age, and may give way, leaving an inch or so with no gasket. This would not leak much, or not at all, if the engine is not running. But once the engine is running, even at an idle, a good bit of oil will get sprayed out through the small opening. The inside of the oil pan is a hurricane of oil droplets. 3. You can have a bad seal at the crankshaft. Its called the front main seal, and along with the rear main seal (at the transmission end of the engine) can go bad and leak. The Neons in 95-98 were known for having seals that didnt hold up. I dont know if they fixed it for 99, as the 97-99 Neons are just about identical. With our 97 Neon we had a leaking rear main seal. It got to where it was dripping a quart of oil out each 75 miles of driving. We drove it from Northern VA to souther Texas and took a few dozen quarts of oil to make it. I cured that leak by pulling the engine out of the car, and replacing both the rear main seal and the front main seals, as well as the camshaft seal (at the upper part of the motor) and putting the engine back in. Now the car gets over 2000 miles to a quart of motor oil - and stays dry underneath, even though it has 160,000 miles on it. I have never heard of a Neon having a crack in the block. Its probably possible, but as far as my experience, usually happens at the same time that the engine blows up from a connnectin rod punching a hole in the block.
As for the gasoline leak at the back. Thats likely to be a bad fuel line hose, or connection at the tank, or it could be a bad connecton between the fuel filler tubing and the fuel tank. If its a bad connection where the fuel filler connects, I suspect it will require dropping the fuel tank to fix.
If the fuel leaks from in front of the tire: Inside the tank there is a fuel pump, fuel sender float, and a fuel pressure regulator, all in one assembly that fits in the tank. They fit into an opening in the tank, which is close to the front side of the rear wheel on the passenger side of the car. The opening in the tank has a rubber gasket, a locking ring to keep it in place, and the hose connections. There are probably 3 hoses. One is the fuel line (fuel under about 49 psi of pressure), a fuel return line from the engine compartment (not much pressure at all), and a fuel vapor line, that lets the fuel in the tank expand and contract with temperature, and keeps the fuel vapors from getting out. Anyone one of the hoses, as well as the rubber gasket for the large opening, and the plastic assembly itself - could be where your leak is. If you run the rear wheels up onto a set of the ramps, you can slide under the passenger side of the car (just in front of the rear wheel) and can see the stuff Im talking about.
Unless the problem is a hose connection that can be replaced from the outside, it may mean taking out the fuel pump/regulator assembly, and putting in a new gasket (about 4" in diameter). This can be done without removing the tank, though it may be necessary to loosen the straps holding the tank up so you can get to things.
Usually you wait until the car is almost out of gas - so the tank isnt so heavy. An empty tank weighs maybe 25 lbs, and each gallon of gasoline adds another 7 lbs.
Sorry to be the bringer of bad news about the oil leak, but if you did a search in this forum, (1st generation Neons) you would likely find postings about the chronic oil leaks on Neons.
The leak at the area where the fuel comes out of the tank is not a common problem, but nothing lasts forever on an older car. Anyone who keeps a Neon running will have to make some of the same repairs that you are looking at. My daughter drives our current Neon. I plan to get another 97-99 Neon, one with a big bad oil leak - since I can pull the engine and fix the leaks - and have a decent car for very little money.
Im looking in the lower part of the U.S. though, so I wont have the rust that happens to cars in the northern states.
If you dont do the work yourself, it wont be cheap. Dropping a fuel tank is probably over $200. Dropping the pan to replace an oil pan gasket would probably be a similar amount - and it wont fix the leak if its a leaking main seal at the crankshaft (or from the camshaft seal at the top of the motor).
By taking the motor out myself and re-installing it, I probably saved at least $800. The three seals I replaced cost about $12 each.
Also....I bought the Neon service manual for the car on ebay. It is the one published by Chrysler, and has all the detail you could ask for. I think it cost me $30. I am retired, and am a self taught mechanic/hobbyist. Its a lot of work to make all the repairs to your own car - but it sure saves lots of money.