You can definitely omit the subject if the first sentence is short enough. In your provided example there is no issue. It's never wrong to include the subject for maximum clarity, though, which would be best in certain contexts like instructions, legal-related, etc.
Your second example is different.
I went out and drank a cup of tea.
This creates an implication that the two things are related - by not repeating the subject pronoun a "blending" effect is happening. "Going out and drinking a cup of tea" would be a single action (consisting of sub-actions) to a typical listener/reader without additional context.
She refused to go out and to drink a cup of tea.
Repeating the to here can sound like you are trying to break that blend/implication, or make it clear that 2 separate things were refused. If you are angry with this person for not going out and drinking a cup of tea, you may say this as an emphasizing device to make her "list of trangressions" bigger. (The ultimate emphasis is created by saying "She refused to go out and she refused to drink a cup of tea!").