First, regarding "ha-adam": Yes, in Hebrew this is the definite article, usually thought equivalent to "the" in English, followed by "adam," which means "man." It is possible to translate this as "the man." However, in Hebrew the definite article serves another purpose: it makes the following noun definite, or proper. In other words, the "the" here may be considered the grammatical function of capitalizing an English word. It is the reason the word is considered a title or name in this context. So "Adam" is an appropriate translation.
Secondly, regarding the order of creation, we know this from other scriptures.
For Adam was first formed, then Eve. (1 Timothy 2:13, KJV)
Because this is later specified so plainly as this, it must have some significance.
Finally, regarding God's "image," it is not to be confused with the concept of "appearance." God's "image" or "likeness" has everything to do with character. Adam and Eve were both created sinless--even as God is sinless. Notably, when, after their sin, they had a child, the Bible records something significant about their son's image.
1 This is the book of the generations of Adam. In the day
that God created man, in the likeness of God made he him; 2
Male and female created he them; and blessed them, and called their
name Adam, in the day when they were created. 3 And Adam
lived an hundred and thirty years, and begat a son in his own
likeness, and after his image; and called his name Seth: (Genesis
The son was not made in God's image. He came in Adam's image.
"Adam," as noted in verse 2, was "their" name. God used the name "Adam" (Man) for humankind, both the male and the female. It was the "male Adam" who gave the "female Adam" the name "Eve"--by which she was afterward known.
Regarding your final question, it is unclear which verses you mean in saying the "second and third" verses, as you only quoted Genesis 1:27. However, the words to which you refer are the pronouns created by adding a pronominal suffix to the Hebrew direct-object marker (DoM). Without a pronominal suffix, it usually just indicates that the next word is the direct object of the verb, and it is not translated. There are occasional usages of the DoM which are translated as "with," but never when it has a pronominal (pronoun) suffix. These pronouns created with the DoM are the equivalent of object pronouns in English, e.g. "him" and "them".