Good afternoon, JK.
(1) Congratulations on being a volunteer ESL teacher. I am sure your students really appreciate your help.
(2) I think your best bet is to check some good grammar books and to look for information on the Web.
(3) I checked my books and found that these verbs take the bare infinitive:
have (American English)
please (as a modal: (you) please close the door)
need (ONLY when a modal, as in "You need not go." Otherwise: You do not need TO go)
help (you may or may not use "to": You help them (to) learn English.
(4) Sorry. I could not find any pattern.
(a) Some are causative verbs (make/ have): I had him cut my hair.
(b) Some are sensory verbs (see, hear, feel).
(c) One famous grammarian observed that the bare infinitive often seems to follow those familiar verbs that we use a lot in conversation.
Best of luck to you.
Credits for this information: Practical English (Michael Swan); The Grammar Book (Celce-Murcia and Larsen-Freeman); A Grammar of Present-Day English (Pence and Emery); Essentials of English Grammar (Jespersen).