Dear Navi,

You wrote:

Which of the lettered interpretations correspond to which of the numbered sentences below:

 1-He was happy to finish the project.

c – however it could be d if the context allowed it

2-He was happy to be finishing the project. –

a – rarely b if the context allowed it

3-He was excited to finish the project.

Same as 1

4-He was excited to be finishing the project.

Same as 2

a-He was happy/excited that he was in the process of finishing the project

b-He was happy/excited that he was going to finish the project in the future

c-He was happy/excited that he had finished the project

d-He would willingly accept to take on the task of finishing the project.

“To be finishing” amounts to a progressive infinitive, so that means continuous or continuing action. Since there is no future infinitive in modern English, #2 could mean b if the context made it clear. Usually, however, the verb would be in the future tense as well: “He will be happy to finish the project tomorrow.”