I can't help thinking that it would be worth your (or one of your colleague's) while to learn some basic SQL, so you can explore the data yourself. It's not that hard if you have that sort of mind. There are various SQL tutorials online, but I don't know which one I would recommend.
Anyway, here are some queries you can try. Note that they are all untested. However they are right in spirit, so should only require minor changes to fix if they fail.
Number of questions of each type in each quiz (if you are using Moodle <= 2.6)
SELECT c.id, c.shortname, c.fullname, quiz.name AS quizname, q.qtype AS question_type, Count(1) AS num_questions
FROM prefix_course c
JOIN prefix_quiz quiz ON quiz.course = c.id
JOIN prefix_quiz_question_instances qqi ON qqi.quiz = quiz.id
JOIN prefix_question q ON q.id = qqi.question
GROUP BY c.id, quiz.name, c.shortname, c.fullname, q.qtype
ORDER BY c.id, quiz.name, Count(1) DESC
The same, if you are using Moodle 2.7+
SELECT c.id, c.shortname, c.fullname, quiz.name AS quizname, q.qtype AS question_type, Count(1) AS num_questions
FROM prefix_course c
JOIN prefix_quiz quiz ON quiz.course = c.id
JOIN prefix_quiz_slots qs ON qs.quizid = quiz.id
JOIN prefix_question q ON q.id = qs.questionid
GROUP BY c.id, quiz.name, c.shortname, c.fullname, q.qtype
ORDER BY c.id, quiz.name, Count(1) DESC
This is the questions used in the quiz definition, so you will see 'random' in the results.
If you want totals for the whole course (not split into quizzes) it would look like:
SELECT c.id, c.shortname, c.fullname, q.qtype AS question_type, Count(1) AS num_questions
FROM prefix_course c
JOIN prefix_quiz quiz ON quiz.course = c.id
JOIN prefix_quiz_slots qs ON qs.quizid = quiz.id
JOIN prefix_question q ON q.id = qs.questionid
GROUP BY c.id, c.shortname, c.fullname, q.qtype
ORDER BY c.id, Count(1) DESC
(I have only given the 2.7+ version of the query. If you need a 2.6 version, you will have to work it out yourself.)
If you want to do it by actual questions attempted instead (so random does onto show up) then it would look like:
SELECT c.id, c.shortname, c.fullname, quiz.name AS quizname, q.qtype AS question_type, Count(1) AS num_attempts_at_question_of_this_type
FROM prefix_course c
JOIN prefix_quiz quiz ON quiz.course = c.id
JOIN prefix_quiz_attempts quiza ON qs.quizid = quiz.id
JOIN prefix_question_attempts qa ON qa.questionusageid = quiza.uniqueid
JOIN prefix_question q ON q.id = qa.questionid
GROUP BY c.id, quiz.name, c.shortname, c.fullname, q.qtype
ORDER BY c.id, quiz.name, Count(1) DESC
There is a third possibility, which is to count the questions in the question bank, but this is probably lease useful (since people may have imported banks of questions, and then only used a few). Also, it is slightly tricky, so I won't try to give it now.