Does SQS reach "unlimited capacity" instantaneously, or do I need to spread messages (ramp up) over some time?

A single Amazon SQS message queue can contain an unlimited number of messages. However, there is a 120,000 quota for the number of inflight messages for a standard queue and 20,000 for a FIFO queue. There is also a throughput quota with SQS - By default, FIFO queues support up to 3,000 messages per second with batching or up to 300 messages per second (300 send, receive, or delete operations per second) without batching. If you require higher throughput, you can enable high throughput mode for FIFO on the Amazon SQS console, which will support up to 30,000 messages per second with batching, or up to 3,000 messages per second without batching. Refer to https://aws.amazon.com/sqs/faqs/ and is a useful resource for additional information about SQS.

Yes it is possible. I have not seen any issues with receiving messages all of the sudden.

There will be a limit to how many messages your producers can reliably send and I would recommend testing this. However, SQS distributes your queue over a number of SQS instances to support scale and resilience.

From AWS Blog

In order to make SQS scalable, durable, and reliable, messages are stored in a fleet that consists of thousands of servers in each AWS Region. Within a region, we save three copies of each message, taking care to distribute the messages across storage nodes and Availability Zones. In addition to this built-in redundant storage, SQS is self-healing and resilient to host failures & network interruptions.

From SQS FAQ

You can scale the amount of traffic you send to Amazon SQS up or down without any configuration

From Basic Amazon SQS architecture

redundantly stores the messages across multiple Amazon SQS servers