Depends on the context. आदमी is a lot like the English word 'man' in that it is sometimes used to mean 'human' or 'person' but is still an explicitly gendered word.
Eg: 'कुत्ता आदमी का सबसे अच्छा दोस्त है'- Dog is man's best friend. Here, you are making a generalised sentence where आदमी does mean human.
'आम आदमी'- Common man. Usually includes women but you can also have sentences like भारत की आम औरत आम आदमी से कम कमाती है (The common woman of India earns less than the common man).
It may also introduce some ambiguity. 'कल हमारे सारे आदमी हड़ताल पर होंगे' - 'Tomorrow, all our men (may mean 'workers' here in this context) will be on strike'. Will the female employees also be on strike or will they be working as usual?
In general, if you are comfortable using the word 'man' in an English sentence, you can probably translate it to आदमी.
With the verbs आना and जाना, the postposition को (corresponding to 'to') is usually dropped. You can think of this being similar to a sentence like 'I am going home' in English. While English drops the preposition for just 'home', Hindi does it for most objects.
But even when it is dropped, the fact that a postposition is supposed to be there in the sentence still influences its grammar. For example, consider the sentence 'पीटर मेरे स्कूल जाता है' (Peter goes to my school'). You can see that the noun phrase 'मेरे स्कूल' is in the oblique case. But for it to be in the oblique case, it has to be the object of a postposition. We resolve this by saying that it is the object of an 'implicit' postposition that's present between मेरे स्कूल and जाता. People in these forums have been referring to this as a 'ghostposition'.
Most words ending in the vowel sound 'ī' (which is what the loop indicates) are feminine but there are quite a few exceptions like आदमी (man), हाथी (elephant), पानी (water) etc which are masculine.
Similarly, most words ending in the vowel sound 'ā' (ा) are masculine but exceptions like महिला (woman/lady), चिड़िया (Bird), माला (garland) exist.
These are only rules of thumb to help you guess the gender of an unknown word and are not actual rules that the language abides by. The grammatical gender of a word is decided by a combination of its meaning, sound, etymology and historical usage and it's impossible to know it for sure just by looking at the word. This is also true for many other languages with a concept of grammatical gender (like Spanish and French).
Persian was the official language of many dynasties that ruled in India at various points of time. So, Hindi has quite a few loanwords from Persian. Arabic loanwords also exist but are much fewer in number. In fact, a lot of Arabic loanwords came into Hindi/Urdu via Persian.