Welcome to the forum,
A glass of water is not an accurate test since smaller volumes of water will warm up much faster than large volumes of water such as your aquarium. Though I would have expected a glass of water to rise to a temperature higher than 13C over night.. What type of thermometer are you using? I would advise to use an aquarium grade thermometer.
Best to fill your tank and begin cycling and then test how warm the water gets. Being in Florida you would likely need to invest in a chiller to keep a consistent water temperature.
Here is some basic information on PH:
"Most municipalities treat their water so that it is within a few degrees of pH 7 (neutral). My local water is 7.2 after treatment. For axolotls, a pH of 6.5 to around 8.0 is acceptable, but 7.4 to 7.6 is probably ideal. pH can affect the toxicity of ammonia and this is discussed below."
"the higher the pH, the more toxic the ammonia. Ammonia can kill, and at a pH of 8 or more, it kills even more effectively. Water temperature can also affect its toxicity, a higher temperature resulting in a higher toxicity. Unless you're extremely proficient with maintaining aquarium systems, a periodic ammonia test is advisable. Even the most experienced hobbyist should occasionally test for ammonia."
Axolotls - Requirements & Water Conditions in Captivity
Ammonia, Nitrite, and NitrAte are part of the Nitrogen Cycle, which is a natural process.
"Axolotls excrete a lot of waste. This is mainly in the form of ammonia (NH3). Ammonia is a very toxic substance in its unionised form, NH3, but this is in constant equilibrium with the ionised form NH4+, and through the process of nitrification, it is converted into the less harmful substance nitrite (NO2-). Other bacteria then convert this into the much more benign substance nitrate (NO3-). These two processes comprise the most important aspects of what is known as biological filtration.
Nitrate is a source of fixed nitrogen (i.e., not Nitrogen gas, N2) for plants and fungi. Live plants, if included in the tank, will consume the nitrate generated from a good filtration system."
Axolotls - Housing in Captivity
You will need a test kit to monitor your nitrogen cycle, because you have to check for levels of Ammonia, Nitrites, and NitrAtes. A lot of users on this forum use and recommend this test kit including myself:
Welcome to API Fishcare: FRESHWATER MASTER TEST KIT These liquid test kits are far more reliable that the test strip versions.
The cycling process can take from 3-6 weeks. For your tank to be cycled, a beneficial bacteria needs to be present to break down the Ammonia and Nitrite into Nitrate.
Ammonia and Nitrite are lethal to Axolotl and are produced from your Axolotls waste and decomposing food. Nitrates are a less toxic by product. During cycling a beneficial bacteria will colonize your tanks surfaces such as substrate, ornaments, filter, and begin to feed on the Ammonia turning it into Nitrite, then another bacteria will develop and begin to break down the Nitrites into Nitrates. The bacteria is known as a biofilter, as it filters the Ammonia and Nitrite into less harmful Nitrate. You can then perform 20-40% water changes to lower your Nitrate when it reaches around 40ppm.
Read these links on the cycling process:
Caudata Culture Articles - Water Quality
Caudata Culture Articles - Cycling
How to Cycle a Fish Tank
It can be quite hard to get to grips with the cycling process but stick at it and ask any questions you need.
I would avoid using gravel full stop to eliminate any chances of ingestion. If you want to use real plants then a substrate of sand about 2 inches thick will be sufficient for the plants to take root, just make sure that you rake the sand during tank maintenance to prevent pockets of anaerobic bacteria building up.
Helpful sites:
Caudata Culture Axolotl Articles
Axolotl Sanctuary
Axolotls: The Fascinating Mexican Axolotl and the Tiger Salamander