Another Main Drain question??? Ideas?

Ok so I've had some issues due to my neighbors with my main drain overflowing/backing up in the past. Eventually a plumber was called out and corrected the problem but once again yesterday I went to flush the toilet and all was well. No back up in the toilet and it flushed as usual. About an hour later,maybe a bit less,I went to change my laundry in the basement and sure enough there was once again sewage backing up into my basement.It started to go down so I started the laundry and went back upstairs to let it drain. Another hour passed and one of my daughters ended up flushing the toilet before I could stop her,(She is almost 3 and rarely flushes the toilet due to the noise scaring her)and when I went back down the water was 3 xs the size in diameter than it was when it initially overflowed. I poured drain-o down it which has done the trick before and sure enough it did start working and then my older girls got home from school and hubby got home from work and the toilet usage sky rocketed and ofcourse everyone forgot that I said not to flush it just yet and half of my basement ended up flooded. It was almost gone by about 9 and everyone went to bed so no more flushing. When it was finally gone about 2 hrs later I flushed the toilet(which was clean) to see if it would back up in the basement again and sure enough it did and now my toilet seems to be clogged as well now. The water gets ready to overflow so I pull up the ball thing in the tank to stop it from soaking the bathroom floor but it still backs up in the basement when flushed. Is there anything I can do besides calling a plumber and using a snake?? Plunging doesn't work,in fact I think it makes matters worse. I do not have a snake and can not get one until this Friday which is pretty much 2 days away to sit with nastiness in my basement everytime the toilet flushes,and I would much rather not call a plumber unless there is no other alternative because I have to go throught he landlord who RARELY answers his phone,and almost never returns calls. Last time I called the emergency maintenance,they had a number to call on the autmoated message(yes automated message for emergency maintenance) and ended up not gettin a return call until 4 days later and then it took another 6 days for a plumber to even be called and show up. My problem with that is,if I had the money to dish out I would call a plumber on my own and pay them there and give the landlord my receipt but knowing him,he will say I should've went through him and now the bill is my problem.Also we live in a double which apparently shares a sewer line or drain line or something because my last overflow problem was due to their negligence but I'm guessing somehow this is our doing this time. I don't know how but if the toilet is now clogged then I'm guessing it is due to our side messing up and not the neighbors(although I may be wrong about that....I have no idea).So if there are any ways to correct this problem before it gets to the Plumber stage I would very much like to hear them and try them out. If Thanks any answers are appreciated.

**** Another thing is my washer does not affect the main drain backing up. I've used it several times and each time it runs(when drain is backing up or not) the water never rises or accumulates. But when the toilet is flushed you can see water come up through the main drain.****