1. For building an API service you do not need Gremlin Server. Gremlin Server has all kinds of features though that might (slightly) relieve the complexity of your service (with the complexity of maintaining Gremlin Server added). The main driver for using Gremlin Server is the support for Gremlin Language Variants, which you do not need.
Resource usage should not differ very much for similar workloads and comparable settings; Gremlin Server requires an additional JVM, but might be more optimized than what you build in house.
2. First check using Gremlin Console for connecting to Gremlin Server. If that works, please report more details about what visualization tool you use.
Best wishes, Marc