"when we have really heavy rain the toilet kind of backs up, the water level starts rising. "
Bad news.
You probably have a combined rain and foul water drainage system at that house, and there is likely an underground blockage or collapse.
Call the company that sends your water bills (or drainage, if different) and tell them what you just said.
If you had some manhole covers you could pry them up with a spade and see which is full. From a semi, the drains at the back of the house often run to the road at the side, in the gap with the adjacent building. Sometimes there is an entire shared run behind a dozen or so houses.
If you are in England and your drains and your neighbour's drain are connected together, then the part after the join, which is shared, is the responsibility of the water company. Any part which carries the waste from only one house is the responsibility of the homeowner.
IME the water co men are often helpful and cheery, and if you give yours a mug of tea and a doughnut, he will sit on the edge of the manhole with sewage flowing over his boots while he eats it and tells you his amusing tales of human waste. They will often poke obstructions free even if they are not shared drains. But if a privately owned drain needs digging up and relating, that's down to the homeowner.
BTW you mention a yard gulley for the sink. I expect this is dark brown glazed earthenware. In pre-1945 house it is pretty sure to be cracked and leaking. This will display by the ground or concrete around it being cracked or sunken, and mended or hollow underneath. A well-recommended small local builder can dig it our and renew. Plumbers are weedy little fellows with petal soft hands and some of them don't like digging drains.