Welcome on our boards!
Is it possible to have triggers on my website. I want to be able to start different conversations (flow entry points) depending on the button pushed.
Yes, this is exactly where we have created event triggers for. This page explains a bit more how event triggers work using our web widget: https://docs.flow.ai/overview/channels/web-chat.html
This is an example what it might look like in the designer:
Can I control the look and feel of the conversation? Rather than a pop up I would like it to be more integrated within the site.
That are precisely the use cases we love to see. We have a JS SDK: https://github.com/flow-ai/flowai-js you can use to create custom solutions (like an AI driven website).
We have built our own web widget using that SDK. Just shout if you have any other questions!