How can I know my travel limits with a weekly or monthly public transport pass when purchased for a specific zone in Frankfurt?

On the page of RVM (Rhein-Main Verkehrsverbund) you can choose between several types of tickets. If you click on "Monatskarte" (month ticket) you will see a link to the stations you can travel to with this ticket.

If you want to travel beyond those limits, you will have to buy Anschlussfahrkarte (connecting ticket).

There are no any discounts to museums etc. as far as I know.

With the daily/weekly/monthly pass for Frankfurt <-> Raunheim you can travel at will in the zones 3700, 5000, 5090, 2645, 3570, 3670, 3675, 6567, 6593, 6649, 6664 as designated in the map below.
This includes notably the zone of "Groß-Gerau", Central Frankfurt and Frankfurt Airport as well as some stops in adjacent areas (of minor importance).

Source: VGF Frankfurt Zone Map

The zone codes above are further explained in more detail in the linked pdf here (in German).

This should answer the first part of your question. The second and third part are already covered in the great answer of @hiergiltdiestfu, how to figure out the zone limits for a particular journey is also covered in their answer and the one of @Neusser.

Your pass will have the start and end zones printed on it and is valid for travel between them. Each zone usually correlates to a municipality boundary which is not very useful for people not from the region, I concede.

To make this all easier to understand, each major zone (those number ends with two zeros) contains smaller zones (those that share the same first two numbers). So for example, Raunheim is located in the major zone 3700 and minor zone 3750. Note that there are a few exceptions this rule (notably Frankfurt Airport) but they shouldn't be relevant for you (as that big exception is covered by your pass anyway).

I will now assume you will buy a regular pass for travel between Frankfurt and Raunheim (without detours). As you can find out on the transport association's homepage such a pass will be valid in the following zones:

3700, 5000, 5090, 2645, 3570, 3670, 3675, 6567, 6593, 6649, 6664

That corresponds to the following whole municipalities (only first column filled out), parts of municipalities (two columns filled out) and stops only (three columns filled out):

To find out in which zone a stop is located you can check the list of zones which contains the municipality (Gemeinde), district (Ortsteil) and zone number (Zielnummer). The municipality is usually part of the stop's name and where relevant (often) the district, too. Note that your pass also needs to be valid for all intermediate stops. You can usually check this by looking at the zone map (complex 5MB PDF).

To make this more clear, if the name of the station starts with "Frankfurt" your pass will cover that. ^{though I suspect someone will state a counterexample in the comments...}

I don't know any reliable and automated way to check whether a specific journey is covered by a pass. In doubt you should ask any public transport employee.

If you want to travel partly outside your pass boundary you can buy a cheaper onward ticket. You find the slightly complex instructions on how to obtain one at a ticket machine at the RMV homepage.

I have never heard of anyone offering rebate for pass holders.