If you can get some "dirty" filter sponges you can pop them into the new filter to cycle the tank a lot faster. Remember to treat any tap water with dechlorinator so not to kill the good bio bugs in the sponges.
No day is a good day for fish advice round my way. The best way to be sure about fish care is to research it yourself....it's better to be empowered with knowledge than relying on people who make a profit from fish meds and replacement fish. Then there's always the bigger tank they make you buy when they sell you an Oscar for a 10 gallon tank.
The favourite saying used by my lfs is "Give them a go...you never know".
Until you get the test kit keep doing regular water changes using declorinator.
Yes they are nerite eggs and I have found some do "wear away" over time but the ones laid in hard to reach spots tend to stay.
I'm not sure of the salinity levels but I have heard accounts of the young being larva rather than baby snails at first with food being very important. They would probably need something like green water, marine micro algae or algae grown in another tank to feed on.