Can't tell you about the prices as I am not from the area but apartments generally do not come with any of those. Of course they have heaters but you have to pay for it. Usually there is a box you can plug a phone / internet modem in as well as a jack for cable TV (though this may depend on the apartment complex you live in), but you have to activate and pay for them.
Keep in mind, most apartments in Germany come completely unfurnished, without a kitchen. So make sure you look for an apartment with Einbauküche (EBK), and sometimes they also come with Übernahme of other appliances like washer / dryer. (Meaning you buy the appliances from the person who lived there before you.) That might be good for you depending on the state of the appliances and the price. Many apartment complexes also have a laundry room where you can use a communal washing machine for cheap.
Air conditioning is almost never built into an apartment but it is not really necessary either. You can get a mobile A/C unit if you like but I've spent all the time that I have lived in Germany (23 out of 25 years) without A/C and I was fine. The weather is usually moderate enough and the buildings are well-insulated so with some common sense you can keep rooms cool during the very hot days (saying days, because usually there are only a couple of weeks of over 30 degrees weather). All apartments come with some kind of heating but it is not usually included in the rent.
As for how to find an apartment, it might be best to try to find some kind of temporary place to stay (like renting a room for a month) and then trying to find an apartment while you're already in Germany. You can use online sites such as
http://www.quoka.de/ or also local newspapers (others might be able to tell you which ones and on which days they publish the new ads).
Something else to keep in mind is that in Germany you do not count bedrooms but bed-/living rooms so a 1 room apartment would consist of one room that functions as a bed and living room (a studio, usually with a separate kitchen and always with a separate bathroom), while a 2 room apartment would consist of a bedroom and a living room (and bathroom and kitchen of course). That is probably what I would go for if you want your mother to come visit you sometimes (a 2 room apartment, 2ZKB = 2 Zimmer + Küche + Bad = 2 rooms + kitchen + bathroom).
You can also do a search on this topic on the forum, there is much general information available about renting an apartment in Germany, and with the exception of prices and (good/bad/cheap/expensive) neighborhoods it's valid all over Germany.