The quote comes from a Times of India news report. As such, it uses Indian English which has some differences from Standard English (the common features of English as used in Britain, the United States, Canada, Australia, New Zealand, and by other native English speakers of European descent). For instance, I don't know what a "history sheeter" is, nor can I work it out from the context. However, that doesn't affect my recognition of the part of speech for "convicted".
I'll answer your second question first because it's easier. Convict is only a verb or a noun, so you can't use it as an adjective.
In English, adjectives usually come before the noun they modify: "a happy man", or they follow a verb after a noun: "a man was happy". There are exceptions to this rule, but it's a good rule of thumb. In this case, "convicted" is only an adjective before the noun.
Now on to your first question: is it an adjective or the past tense form of the verb "to convict"? The sentence can be simplified down to:
A history sheeter was released.
Everything before "was" is a clause acting as the subject:
A history sheeter convicted in 2013 for raping a five-year-old girl in the neighbourhood
This can be illustrated by using two sentences:
The next sentence is about a history sheeter convicted in 2013 for raping a five-year-old girl in the neighbourhood. He was released in 2018 early for good behaviour in jail.
Now it is clear that "a history sheeter convicted in 2013 for raping a five-year-old girl in the neighbourhood" is not an independent sentence, but a sub-clause that can be replaced by the pronoun "he" once it is in context.
It means "a history sheeter who was convicted ...", with "who was" removed by ellipsis. So, the entire verb phrase is "was convicted". "Convicted" is the past participle of "to convict", which is identical to the simple past tense form, but two conjugated forms can't appear without something else in between.
The sub clause "[who was] convicted in 2013 for raping a five-year-old girl in the neighbourhood" is acting like an adjective here, but appearing after the noun.
So, why can't "convicted" be an adjective the same way as "I was contented yesterday"? This is because "to be verbed" is a passive construction. The man was convicted by someone else; someone convicted the man. Once someone has been convicted, they can't be "unconvicted". The Oxford definition of "convicted" is:
Having been declared guilty of a criminal offence by the verdict of a jury or the decision of a judge.
It's like having a broken leg. The leg can't break itself, some external force must do it. Even after the leg has healed, it was still broken at some point in the past. So the man can't be both convicted in 2013, and not convicted afterwards, which the adjective would suggest if it was used.
Many adjectives do have exactly the same form as the past particle of many verbs, so it can be tricky to identify which is which. You need to identify the underlying grammar ( which may be difficult because of ellipsis or other idiomatic grammar) and also the meaning and use of the word in question. You won't be able to tell whether a different word is an adjective or a past participle by the grammar alone.