1. The house is close to the river.
2. The house is near the river.
3. the house is by the river.
4. The house is next to the river.
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Are they all the same and grammatical? Do we have some more similar expressions?
All are correct and all are used. We might also say, "The house is on the river", which is technically incorrect (grammatically correct, but the house is presumably not floating). It is commonly used, though.
Yes, all four sentences are grammatically correct and convey a similar meaning. They all indicate the proximity of the house in relation to the river.
Other similar expressions that can be used to describe the proximity between a house and a river include:
1. The house is adjacent to the river.
2. The house is alongside the river.
3. The house is within walking distance of the river.
4. The house is in the vicinity of the river.
5. The house is a stone's throw away from the river.
6. The house is within sight of the river.
7. The house is just a short distance from the river.
These expressions all convey the idea that the house is located in close proximity to the river, though the specific distance may vary.