It's great that you're taking the lead and refining your ideas to come up with a well-defined research proposal. Your prospective advisors will appreciate the effort and it will likely increase the chances of them accepting you as their student.
Cost is certainly an important aspect in determining the feasibility of a project. From your question, it appears that the main cost will be for computing power. It would be a good idea to attempt to estimate computational requirements to the best of your ability, by looking at relevant literature that you base your ideas on. If the costs are likely to be extremely high (as an extreme example, take the cost of training GPT-3), you could shelve the idea and think of something else.
However, no matter how carefully you go through this, it is always possible to miss something vital. Particularly, since you are undergraduate and it sounds like you are not yet very experienced in this field, you may miss factors to consider that are obvious to your professor. This is completely fine: you learn by working on projects, and it is the role of the supervisor to guide you through it. As long as you make a good faith effort and are enthusiastic to learn, it is very unlikely that they'll think you are an "idiot" because of this.
So, it would be a good idea to prepare well, but not wait too long to approach the professor. Their inputs can prevent you from wasting time on infeasible ideas, and also help improve your proposal. If they consider the idea feasible and interesting and accept your project, they would very likely cover most of the costs too by providing access to their lab's resources, so external funding, while welcome, may not be necessary.
Referring to your second question: yes, you can certainly discuss other ideas and ask for suggestions, or offer to join an existing project. It is unlikely to be unusual for professors to provide complete project ideas at the undergraduate level.