(1) Since Greek only has one present tense, "believes" is probably a better translation than your six-word phrase, which added all kinds of things to the text. In English, "believes" also has a continuous meaning, anyway.
The Greek present tense has little to do with the English present tense! They share the same name, but are both significantly and radically different! The English present tense on its own simply indicates something that is occuring in the present. The Greek present tense indicates something that starts to happen, and continues onwards into the indefinite future. The event may start happening in the past, in the present, or in the future - the Greek present tense has no indication of time! When the Greek present tense is combined with the indicative mood, then it implies something that starts happening in the present, and extends into the indefinite future. πιστεύων is in the Greek present tense, and it is not in the indicative mood; it only indicates ongoing action.
The English word "believes" on its own does hint of an ongoing or continuous meaning, but generally speaking, and in most people's eyes, I would say it has the more common meaning of a one-time event. That is why adding "
continues to believe" or "
keeps on believing" clarifies this tense and mood; perhaps the modifiers should be in italics like the KJV used for interpolated words.
(2) You're misunderstanding the subjunctive here. It does not say "whoever believes in him might perhaps have eternal life." Rather, with ἵνα the subjunctive simply expresses purpose: God gave his one and only Son so that ...

The fact that ἵνα is a conjunction has little to do with the subjunctive mood of ἔχῃ. ἵνα is simply used to join the two thoughts together. How does it modify ἔχῃ? ἔχῃ is clearly subjunctive, and it implies uncertainty. Any interlinear or ultra-literal translation will show this is true:
YLT: for God did so love the world, that His Son -- the only begotten -- He gave, that every one who is believing in him may not perish, but
may have life age-during.
Concordant: for God did so love the world, that His Son -- the only begotten -- He gave, that every one who is believing in him may not perish, but
may have life age-during.
ABPI: thus indeed loved - God the world that the Son the only begotten he gave that everyone who believes on him not might perish but
might have life eternal
(3) You're mistranslating the verb: πιστεύων εἰς means "believing in." Let's look at what some Bible versions have:
ESV: For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.
NIV: For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.
HCSB: For God loved the world in this way: He gave His One and Only Son, so that everyone who believes in Him will not perish but have eternal life.
NLT: For God loved the world so much that he gave his one and only Son, so that everyone who believes in him will not perish but have eternal life.
MESSAGE: This is how much God loved the world: He gave his Son, his one and only Son. And this is why: so that no one need be destroyed; by believing in him, anyone can have a whole and lasting life.
They all have some version of the literal translation I gave.
Thanks, but those are not literal enough for me. If I may quote some of my sources again:
YLT: for God did so love the world, that His Son -- the only begotten -- He gave, that every one who
is believing in him may not perish, but may have life age-during.
Concordant: for God did so love the world, that His Son -- the only begotten -- He gave, that every one who
is believing in him may not perish, but may have life age-during.
Most popular bible translations have no incentive to clarify this verse to the n'th degree as the YLT and Concordant has done, as I imagine that they have sale numbers clearly in mind. A translation which changes this well-known and popular verse will likely stay unknown and unpopular. It is hard to go against 400+ years of tradition!
Secondly, check the scholarly lexicons for the meaning of pisteuo ... Liddels include comply, obey; TDNT includes trust, obey. The opposite of pisteuo - apeitheo - clearly means "disobedience" or "refusal to obey" (1Pet 2:7), which confirms the additional meaning(s) of pisteuo as I've translated it - "trust" and "faithfully obedient".
I stand by my translation.
"For Elohim so loved the world, that He gave His only begotten Son, that whosoever
continues to be faithfully obedient in Him
should not perish, but
should have everlasting life." Jn 3:16