No to both questions. Floating plants are rapid growing, and they will show signs of settling and growth fairly soon. Once this is obvious, you are ready to add fish. Floating plants will take up so much ammonia they are often termed "ammonia sinks." As this is a new method for you, it is safe to be cautious. I can set up a new tank in a day and add fish immediately, but I've been doing this for over 30 years, and I have growing plants from other tanks to use, and to be honest, I know what I am doing and I can recognize the signs should anything go awry.
The pH is interactive with the GH (general hardness, which is basically the level of dissolved calcium and magnesium in the water) and KH (carbonate hardness or alkalinity). These serve to regulate or buffer the pH, and their impact depends upon their own levels. You cannot usually alter the pH without altering the GH/KH. This can get very complicated, and if fish are in the tank, deadly. You should find out the GH, KH and pH of your tap water which you will be using; then select fish that are suited to that water. This data may be on your water authority's website. There is really no need for a GH/KH test kit as once you know the level it is not going to change. The pH is different; depending upon the buffering capacity of the GH/KH, the pH may fluctuate or be a warning sign of deteriorating conditions, so a pH test kit is a good idea. [There is much more to this topic.]
Plants have some effect on GH and pH, but to explain that here would I think cause more confusion. This is not a problem anyway, as it occurs in natural habitat waters and fish are well able to handle it. As for plant fertilizer, this may be advisable. The more plants you have, the more nutrients they will need, in balance with the light over the tank, and the fish being fed are unlikely to provide all of this. A comprehensive (complete) fertilizer is what you want; there is no benefit in adding individual mineral nutrients, this can create all sorts of problems. And the GH of your source (tap) water provides some minerals and we need to know that. I don't know which country you are in, so I won't recommend products now.
No, unless you select a sponge filter which connects to an air pump. The filter should provide surface disturbance around it, and this plus the movement of water through the tank--both of which are the job of the filter--are all you need.
