Could a producer survive without a consumer

In theory they can survive, however, with nothing to challenge growth, producers will expand till they become sources of competition for each other. With consumers certain producers will thrive because other producers will be consumed by consumers. If there are no consumers the producers will compete with each other and growth will always be highly limited.
If there are consumers some producers will be consumed but others will thrive due to lack of competition from other producers.

Yes and no. A producer can receive energy from the sun through photosynthesis, so it does not necessarily need a consumer. However, consumers help balance the population so that producers do not run out of space or nutrients. In addition, after being consumed, primary consumers will absorb nutrients and energy and pass them down to the next level of consumers. Nutrients will be returned to the atmosphere once the consumer dies (called nutrient cycling) and can be used again by the producers.