Welcome to the Forum ! As you will see, homework questions that are simply posted without showing some effort are usually roundly mocked.
Your questions are vague enough that they might not be from a PLC class. I'm curious to find out.
1. How many analog inputs can the slc500 control ?
Trick question ! A PLC controls its outputs, not its inputs. Refer to the syllogism in Bernie Carlton's signature.
The Input data table of an SLC-500 is 256 16-bit words long, so in theory you could connect a large collection of analog input modules. If I were answering this on a test, I would say "the number of analog input channels is limited only by controller memory".
2. Why can't a program be downloaded to a PLC when it is in run mode ?
That's just the way they work. PLC's typically have pre-allocated memory and resources, without enough to load a second program in parallel with the first. Most PLCs can have smaller program edits done while in run mode, but not whole downloads.
3. What is the purpose of a watchdog timer in a PLC ?
The goal of a PLC is to rapidly solve control equations by reacting to inputs and communications. If the user program takes too long to execute (or there is a malfunction of the controller operating system) the watchdog is usually a low-level OS function or separate circuit that interrupts the system and shuts it down if the user program takes too long to execute. A typical PLC watchdog is in the 500 to 1000 millisecond range.
4. What is the largest positive and negative floating number point number that can be specified in a SLC-500 PLC.
SLC-500 controllers, like many other controllers, use a 32-bit Floating Point storage mechanism, called IEEE 754 Single Precision Floating Point. The maximum positive and negative values that can be represented by IEEE 32-bit Floating Point are about +/- 10^38.53.
5. Why can a program be executed when the program is on remote program mode ?
It cannot.
The "Remote" position of the SLC-500 keyswitch (and the corresponding operating mode) just means that a remote programming terminal can switch the controller into RUN or PROG. The SLC-500 program does not execute when the controller is in Remote Program or Program mode.
6. What is the maximum amount of up and down counters the slc500 can have?
Again, this depends on controller memory. A single Counter or Timer data table supports up to 256 Counters or Timers each, and you can add Counter or Timer data tables until you're out of controller memory. There is no pre-set limit on the number of Counters or Timers in the SLC-500 controller.