There's clearly some misunderstanding here, but I'm not sure if it's a misunderstanding of the game rules or just how probability works in general. You may be thinking in terms of the Gambler's Fallacy -- if you flip a totally fair coin and it comes up heads six times in a row, the chance of getting heads on the next flip is still 50%. It's not "due" for a tails or anything. (If you flipped heads six times in a row on an actual coin, there's a higher chance that the next flip is also heads, because it's increasingly likely that the coin isn't actually fair -- the weighting of the coin or the way you flip it might be causing it to prefer heads -- but the difference would have to be pretty large to have an impact that dramatic.)
Generally, when the game rules refer to a percent chance, you should read that as rolling d100 ('percentile'). The event happens if you roll equal to or lower than the stated number. You can use any fair method of generating numbers, but d100 is usually the simplest way. In this case I would tend to have the caster roll for themself, but the DM could certainly roll instead if they wanted to, since the spell doesn't actually specify.
Each time you cast wish to do one of the 'other' effects, you roll 1d100, and if you get a 33 or lower, you lose the ability to use wish ever again. In theory, you could use wish hundreds of times, as long as you continue to roll over 33 each time. There's no guarantee of failure on roll #34, either by the game rules or by the physics of how dice operate -- so no, the caster isn't bound to eventually roll poorly and lose wish for good.
Only one caster in six will get past their fifth use of wish without losing it forever, though, and only one in a hundred will pass the tenth roll successfully. Getting past the 32nd roll is a trick only one caster in 600,000 can pull off, and managing to wish fifty times in a row without losing the spell is a one-in-a-billion chance. But it can still happen...
As to the other questions, 1-2 on 1d6 is pretty close, but would have a minor deviation of a third of a percent, as you identified, so I wouldn't do that, myself; I'd roll an actual percentile. That said, the difference is so small it probably doesn't matter.
I don't know of any spells, feats, or abilities that could change a percentile roll that's part of a specific spell; did you have something specific in mind? Mostly roll adjustment abilities like Lucky say what they can affect, like 'attack rolls, saving throws, or ability checks', and this roll is none of those. There are a few abilities that can change a percentile roll, but the ability applies to a single, specific kind of roll (like the Wild Magic Sorcerer eventually getting to roll twice on the Surge chart and pick one of the two results). They wouldn't apply to a wish.