The anus and the vagina are not considered beit hastarim; that category applies to external folds of the body such as under the armpits, under the breasts, between the thighs, etc. The anus and vagina are viewed as more closed and anything within them is considered balua ("swallowed up"). Therefore, you do not have to insert anything into either the anus or vagina to clean them - just wash the outside area with soap and water as you normally would.

There is a custom to use the bathroom prior to immersion in order to assure that one is not "holding in" either urine or stool, which might interfere with a relaxed immersion. You may choose to relieve yourself prior to bathing or afterwards. In that case, you can wash the external area again with a wipe or washcloth.

Note that the mouth is considered an area that, while somewhat internal, is open to the outside and needs to be properly cleaned (e.g., teeth brushed and food removed from between them).

The only thing that needs to be inserted into the vagina are the bedikah cloths for the required number of bedikot. You are correct that superfluous exams may lead to problems (scratches for example) or questions and therefore should be avoided.