How can I detect the amount of memory slots I have?

How about going by command-line without any third party installer?

wmic MEMORYCHIP get banklabel, capacity, caption, devicelocator, partnumber

gives you something like this

BankLabel  Capacity    Caption          DeviceLocator   PartNumber
BANK 2     4294967296  Physical Memory  ChannelB-DIMM0  xxxxxxxxxx-PB

CPU-Z offers detailed information about RAM slots, timings, XMP profiles, etc. As for max supported you normally will have to look up the limits imposed by your northbridge.

To answer question (1) using WMIC:

wmic memphysical get MaxCapacity, MemoryDevices

produces something like this:

MaxCapacity  MemoryDevices
16777216     2