In Mexico, the general rule is, if there is a lined wastebasket next to the toilet, always put your used tp in there. This does not apply to all hotels and many AIs, as they have their own separate septic systems.
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Under no circumstances should you ever flush tampons or pads. They will not break down quickly enough and can cause serious blockages.
Additionally, toilet paper can take years to break down and decompose. So, flushing is the more environmentally friendly option. You could try an alternative solution, which may be using recycled toilet paper if you can find a brand that is BPA-free, chlorine-free, 100% recycled, and unbleached.
As a precaution you should not drink tap water in Mexico even if it's purified at the source. Most resorts and hotels provide bottled water or large jugs of purified water for you to refill your bottle. Velas Resorts stocks its suites with bottled water daily as included in it's all inclusive plan.
Anything that can be sanitized or laundered and used by another guest is the hotel's property and is considered stolen if you take it. They aren't going to do anything if you take toilet paper, kleenex, or coffee packets, although you have to be pretty cheap to do that.