If your saxophone is leaking spit, can it damage the pads?

You should not be able to see fluid, be it spit or condensation, coming out of the upper holes. I wonder whether the pads are closing correctly.

If in fact you feel air escaping from your lips (not going down the saxophone "tube"), then you have embouchure problems - might benefit from lessons and/or a softer reed.

Similarly, if screws are working loose, take the sax back to the rental place and tell them to do a proper overhaul (cleaning the screws and rods so they won't work free, levellng pads, etc). You're paying good money right now for a poor axe.

According to Charles Harris, who has been playing saxophone 35 years and been repairing them since 25 years, he has answered this question in a relative way on qoura.com

You can read the whole thing here:

https://www.quora.com/How-damaging-is-spit-to-a-saxophone-over-time

Hope this helps!

As said in the comments, what you observe is likely to be condensation, which is a normal phenomenon on a sax.

However, it is a good habit to clean it with a pad saver after you played in order to remove the moisture from inside the sax, preventing corrosion and eventual pad damaging… After cleaning with the pad saver, do not let the saver inside the sax as it will retain humidity inside the sax and prevent it to dry properly…

If you have the appropriate screw driver, fastening a bit these screws may prevent you from a bad surprise, like a bar falling off or something like that ;).