When used on their own, and are always transitive. However, when used as an _auxiliary_ verb, and will always be used instead of their intransitive counterparts. The main verb, the verb that this helping verb attaches to, is the real determiner of transitivity. Transitivity has nothing to do with the auxiliary verb. For example, because is intransitive, will also be intransitive. On the other hand, is transitive, so is also transitive. The transitivity between the main verb and the auxiliary does not need to correspond.
and cannot be used as auxiliary verbs. It is just ungrammatical. and are used every time, as stated above.
As for , that is its own word and is arguably separate from the verb , and it can definitely be used where the verb would typically be intransitive. However, is also a valid word for "beginning", and the difference between the two is somewhat subtle and tough to explain (but I found an explanation here).