1. Can you tell me which words are right for the sentences below?

I am  riding/taking  a bus. -- Both are OK; taking is more common/natural for mode of transportation.

I  take/ride  an airplane -- Ditto

I waited at the bus stop for half an hour/half hour. -- Half an hour or a half hour

All  is/are fine. -- Both are OK.  I checked my puppies and all are fine.  I checked my health and all is fine.

2. Why the sentences below do not have articles for the nouns? What is the reason?

Vehicle is the general word for ... -- No article because vehicle is not being used syntactically; it represents itself only, much as I use it in this sentence.

You can use vehicle to refer to something that you use in order to achieve a particular purpose. -- Ditto

3. When you are writing in bullet form, can you do away with punctuation marks too (like here)?-- I see no bullet form written here, but punctuation is normally required.

A conference for students in pottery (no comma ?????) sponsored by Pottery Association -- Yes, a comma is needed before sponsored.