As for the outlets as someone else already explained, it's 1875w total not each outlet. Also, like someone has already said, it's all about amps. Most bedrooms in apartments or townhomes have a 15 amp rating, you can easily find this out by finding the breaker box which should be somewhere in your apartment. Most individual breakers are already marked for which room or area of the home it supports, if not just shut them all off and turn one on at a time and find out what room or area it's powering. Most bathrooms have 20 amp breakers. So if you need the amps you will need to run a dedicated extension cord that is rated for the watts you need, from the bathroom to the grow room.
An "example" of totaling up how much amps you're using is: two 600 watt digitals uses 5 each for a total of 10 amps, inline exhaust fan uses 1.5 amps, two regular fans use .25 each for a total of .5 amps. That's 12 amps total being drawn from that room, the maximum is 15 amps because the breaker has a 15 amp circuit. Now one could add a few more electrical items to that room, but honestly, you never want to draw the maximum amps so 12 would be a "safe" maximum on a 15 amp circuit. And you can draw all 12 amps from one outlet or all of them in that room on that circuit, it doesn't matter.
As for the landlord receiving copies or a record of every monthly bill, it's amount and usage stats I don't know. Most do not, most have it setup to where they are just notified if the bill isn't paid on time, and this is usually just with city utilities. Example being if you had to pay for sewer and water which is usually a quarterly bil, many landlords have it setup to where if you don't pay it, they are notified and they or the rental agency that many owners use to do all the work for them will send you a pay or vacate notice, each state has it's own time limits. But for a landlord to keep track of a regular electric and gas account which he is not responsible for is not the norm, unless it's in his name it's not an issue. At most he may be notified if it isn't paid. What you need to worry about is inspections or visits from the landlord or a maintenance person. Make sure the room you're using as a grow room has no "mains" meaning no circuit box, no water shut off switches, connected to the furnace room, has the attic access, ect. Have a plan for when an unexpected visit happens, and if it's like most apartment knobs it can lock from the inside and you can pop the lock with a paper clip or small screwdriver by poking in the small hole on the knob. Always secure that room and prepare fro an excuse for not being able to go in there is someone asks to. When I lived in a apartment I had a Ebay store so I had tons of boxes and I strategically divided my grow room with them and other shipping supplies so if someone peaked in it looked as if the boxes filled the entire room. Also, average out all the electric bills since you've lived there. If the total say is 650 kWh(kilowatt hours) each month which is a modest amount for a single person living alone, never exceed x2 that amount. For veg say if you use a 400 watt, stop using the dishwasher, or fill your empty fridge with food or gallons of water so it runs less, put in a few CFL or LED light bulbs. Try to even out that average. Then when flower comes around, stop using the dryer if you have one, use cold/cold cycle on washer, turn down water heater, install all CFL and LEDs, use your pc 50% less, turn lights off, use appliances less, quit using oven if electric, ect, ect. The key is to gradually raise your usage to the point it's not noticeable.
You HAVE TO have a carbon filter for odor, maybe use the exhaust of those rooms to partially heat the apartment if you live in a cold climate. In hot months if you have central air you can cut off the main rooms from receivng that air so only the grow room and the bedroom where you sleep will have cool air. In apartments it's all about sacrifice and saving energy so you can transfer it to your grow room.
Expect the worst and be prepared for it and you'll get through it if it gets hairy. Don't be paranoid about the bill, I know people that went from 150 a month to over x2 at 300-400 and as long as the bill is paid it's all good in most cases. Peace