How can I do api calls in the custom node created with execute type

Questions
2

You can simply use this.helpers.request(options);

All the code of all the nodes is on Github. One simple example which shows how you can achieve that is this one:
https://github.com/n8n-io/n8n/blob/master/packages/nodes-base/nodes/LinkFish/LinkFish.node.ts#L298