What is the ruling on asking many people to complete a questionnaire for a university research project, and informing them that everyone who volunteers will be entered in a draw for 1 small cash gift? What is the ruling on giving everyone who completes the questionnaire a token gift (e.g., pencil)?

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
There is no objection in collecting certain questionnaires for some legal purposes if there is no violation of Islamic rules.
Therefore, there is no harm according to Shariah in presenting nominal gifts to encourage people to complete the questionnaires. This practice is not considered as gambling since those who participate in the research do not pay fees. Gambling means one may profit or may lose depending on chance. But in this case, people write something that they have been asked and then they get a prize for which they paid no fee so, it is permissible in Shariah.
Allah knows best.