Learn English – helping verbs in the past tense

Good question. The answer is that irregular means 'not following the rules'.

In this case, the "rules" are for the regular verbs, and they say

1. The past form and the participle form of regular verbs are the same
   (i.e, "2nd form" = "3rd form")
2. These forms of regular verbs add "-ed"
   (sometimes spelt differently).

So irregular verbs must violate those rules, which means either

1. they don't use "-ed", or
2. they have three forms instead of two.

Any dictionary will give the forms of a verb, but only in code. Here's the key:

1. if the dictionary indicates that the past form is "-ed", or "-d" or "-t",
   or if it's a special form like spent, where "d" changes to t,
   or if there is no participle form given,
   then it's regular
2. if the dictionary gives three forms (like sing, sang, sung),
   or no verb form ends in "d" or "t" (like wear, wore, worn),
   then it's irregular.

Plus, there are very few irregular verbs, and they follow common patterns, and they occur often.
Which means you can learn them from a list, like this one.

"He went home. But before that he had eaten a sandwich." - the most technically correct answer - you use past perfect tense in its intended purpose.

"He went home. But before that he *has been eating a sandwich." - incorrect by the grammar rules, and I don't think present perfect continuous tense is very common in casual speech in this case. I'd use past perfect continuous ("...he had been eating a sandwich").

"I lay in my bed and remembered the last dream I *had have." - I think the tense is right (past perfect again), but you got the past participle wrong - it should be "...I had had".

"I lay in my bed and remembered the last dream I *have had." - see the second case.

In general, it's also acceptable to use Simple Past in place of Past Perfect if the order of the events is clear - so "He went home. But before that, he ate a sandwich" and "I lay in my bed and remembered the last dream I had" would also be correct.