Is the Bachelorette scripted?

While I can't say for certain if your local version of The Bachelorette is scripted or not, considering that it's the same kind of series as The Bachelor Australia and given what happened in Season 2 of it I would suspect that things are not as what you expect them to be of an unscripted series. Citing Wikipedia:

Garvey proposed to Frost in South Africa, which she accepted. On 6 October 2014, it was reported that Garvey had ended the relationship and that he and Frost were no longer engaged, with Frost stating she was "completely blindsided" by his decision. During an interview with Carrie Bickmore on The Project, Garvey declined to answer whether or not he was involved with anyone else, however on 10 October 2014, it was reported that Garvey was in a relationship with second runner-up Louise Pillidge. Garvey confirmed to Woman's Day that he and Pillidge were dating and that he had broken up with Frost two weeks before the finale aired, while Frost eventually went on to star in the first season of The Bachelorette Australia. Garvey and Pillidge announced their breakup on 18 April 2016, after 18 months together.

This at least tells us that you're first point may be correct as in real life at the time the finale aired Blake and Sam had already been dating and broke up rather than having just gone though the proposal. Also the TVTonight link of Citation 17 also says:

“It highlighted that Sam is someone I absolutely love and care for but not in the way of a romantic partner. I realised there was a change over the six weeks – the chemistry changed.

...

Viewers who watched a proposal air on Thursday night’s finale will curious about when the relationship ended.

It raises questions about whether the finale should have edited out the proposal and simply left it with his choice of winner, which would have been a fairer account of events.

This also would indicate the time lag between production and airing.

Also in the TVTonight link of citation 18 says

One bachelorette, who has chosen to remain anonymous, has spoken to New Idea and revealed that the producers were “pushing him to pick Sam”, and he had let Louise go as he was worried she wouldn’t be able to handle the pressure of being in the final two.

“After the show finished airing, Blake rang Sam and told her he thought he made the wrong decision, that he thought he should have chosen Louise, and that production pushed him to choose Sam!” says the source.

This shows that Production pushed for a certain choice, though I have no idea way it shows that production goes against the perceived premise of the show in that it's the Bachelor's lone choice of who he chooses to keep and let go. (It's either that or Blake is a scumbag who's blaming the production team for his mistake rather than owning up)

2. How would the girlfriend know where to find the production crew within 1-2 days of watching her ex appear on the previous week's episode?

Location wise it might not be all that hard to work out. From what I have seen of both The Bachelor and The Bachelorette there is multiple limousines going to the same place in what is aired as the first night where the Bachelor/Bachelorette meets their prospective partners. Aside from that you could watch out for the TV Crews during dates at famous places and try and following their route on the way back (but ofcause you have to be really fanatical about this to pull it off). And if the set looks the same you might be able to look on Google Earth for the similar outdoor scenery

That side if we assume the same Time Lag with The Bachelor Australia Season 2 to the season of The Bachelorette you refer to it is impossible for anyone to see the episode air then appear on the show the next week since that next week as already been shot and edited

3. Why would someone expose themselves as a two-timer on national tv? That is dating life suicide. If someone does a check on him online, they'll see articles of his infidelity and he'll never get a date ever for the rest of his life.

Assuming that it's true you're assuming that someone would Google another before dating. I remember that in a previous season of Big Brother in Australia one of the housemates was kicked off the show for turkey slapping another contestant (NSFW WARNING: Urban Dictionary link to the definition of a sexual act). I don't recall who it is but only that it was unwanted and if i was a woman i wouldn't want to date a guy¹ who would do that without my consent but i wouldn't think to Google up a date's name to check if they were on Big Brother (unless they mentioned it but I doubt it unless they have an excuse lined up).

You would also want to consider if the production team knows everyone about a certain person as people can be different when they know they have a camera on them. While one would assume contestants are interviewed and are told to be truthful during this process there's nothing to prove they are until they are caught out or further background checks are done which may or may not lead to some sort of reprimand (in one season of Big Brother in Australia 3 housemates lied about them not being Single and were put up for eviction). It's also possible the person in question didn't see it as a big deal anyway.

Considering how the show you talk about is The Bachelorette and the incident you mention was about one of the many Bachelors in The Bachelorette Australia there was an incident of sorts which would make one think less of one of the Bachelors on there. While I can't remember who it was during an episode of Season 1 of The Bachelorette Australia before the Rose Ceremony one of the guys pegged Sam as being attracted to money claiming that all the guys knew what Sam really wanted (alot of them were very successful people). However after offending her during their one-on-one time and the other guys with his comments he attempted to interrupt Sam's one-on-one time with another Bachelor to try and explain away and play down his comments, even trying to do it after the rose ceremony where he didn't get a rose.

This was still early in the show so on the one hand one can think less of the Bachelor because he made such assumption without getting to know Sam more. On the other hand it does make for great TV (I general don't watch the show but I paused my game just to see it play out after seeing the ads) regardless if said Bachelor was genuinely making snap judgement on Sam's character or if he was planted there by Production and told to do certain things (think about other reality TV shows which have "villain" contestants, are they really like that or just playing the role to add conflict to the series)

Now while I am critical of The Bachelor Australia and shows like it, applying it to the version of The Bachelorette you talk about it doesn't mean to say the success stories at the end are fake and it's possible that some seasons may actually be less fake (producers trying to push for certain decision, the time lag between production and airing).

If the season as ended and/or any Bachelor/Bachelorette get interviewed look out for them saying that they had to keep the results/events a secret until an episode aired or them saying they saw the episode of themselves on TV. This would indicate the time lag between production and airing and if there is one it would indicate that your second point is valid

_{1: If I was a woman I still wouldn't date a guy anyway}