hi,
1. They are happy to go to beijing
but
2. they are happy to attend the meeting
all the other examples you have given mean the same.
the first one is different because, it has two main verbs.
They go to Beijing and attend the meeting.
Use adverb i.e. happily for the verb which is making them happy
the rule is rather simple as above. The adverb must be near the verb which it is describing. because as you see in the name of "adverb" it should be added to the verb.
In case of sentences with one verb, the positioning doesnt matter but be careful and place the adverb just infront or after the verb sentences with multiple verbs
good luck :)