Multiple lines of independent evidence. The Sun's internal dynamics come from our understanding of everything from the largest Einsteinian space-time warping to the smallest quantum tunnelling (the latter finally explaining the Sun's emission spectrum, something classical mechanics never could). The Earth's age comes from radiometry, geology, cosmology, etc.
Right, the sun's age is determined by multiple independent lines, however, the age of the earth is determined by evidence and a certain amount of assumption. Both the sun and earth are aged assuming the Nebular theory which is widely accepted but has some problems.
First, what evidence do we have that the sun formed first? The evidence for the sun's age is around the 4.6 billion year range. The earth's 4.57 but this is based on comparing meteorites and other fossil evidence. There is no precise method of dating the earth.
They could... but it's unlikely. The sheer abundance of independent bodies of evidence means that the probability that the Earth is indeed 4.54 billion years old is exceedingly high - not just 'best guess', but 'almost certainly the right figure'.
Abundance of independent bodies of evidence for the age of the earth? You mean meteorites and fossils?
The data come from two different angles, one lowering the maximum age (it can't be older than 4.54 billion) and one lowering the minimum age (it can't be younger than 4.54 billion). That all this evidence hones in on exactly the same value is called consilience, and it makes the evidence is phenomenally stronger and almost impossible to budge.
Why can't it be older than 4.54? There are varying dates as high as 4.57 some even claim 4.6.
Isaac Asimov penned a famous essay called The Relativity of Wrong which explains quite nicely the error you're making. Basically, just because scientific conclusions change doesn't mean that they change completely. We've gone from a flat Earth to a spherical Earth, but that doesn't mean we'll next say the Earth is cubical, and then that it's dodecahedral, and then that it's toroidal. Scientific ocnclusions don't jump around willy-nilly, they refine.
I never claimed it was willy-nilly. However, jumping 100 million years exemplifies the imperfection of fossil evidence.
Even great paradigm shifts follow this trend: the changeover from classical to quantum mechanics didn't radically alter the notion that heavy things fall or snooker balls will collide. It refined the small-scale behaviour, but didn't overturn the major conclusions drawn from the old theory (heliocentrism, escape velocities, accretion disks, etc).
Ok?
So while, yes, it's entirely possible that all our observations and measurements are all erroneous and just coincidentally hone in on exactly the same value, this is bewilderingly unlikely.
When one makes the assumption for instance, Nebular Theory and base evidence using that model it would come out at the same value. It would not be erroneous nor coincidental. The observations and measurements are using base assumptions. For instance, if you see meteorites that date to 4.67 billion years old and earth rocks that date 4.5 billion years old or there about and assume that the sun must be older than that using the Nebular Theory it would be based on evidence and not be considered erroneous or coincidental.
This simply untrue. The consensus has been that the Earth is 4.5 billion years old, and has been this way for decades. What the first lifeform was is still under debate, but only a handful of hypothesis dominate the discussion. Whether bare RNA or polymers in a micelle bubble, we're looking at very primitive biology - it's not going to be a leprechaun. Even in uncertainty, we know the kinds of things it's going to be.
I am not claiming that they would find a rose dated at 5 billion years old. However, the first life form of plants could be present very early on that lead to flowering plants. We also know that there is plenty of time for life to have existed on earth and been destroyed. So there is that possibility as well.
Yes. So? There are limits as to how early life can arise - you won't find primates prior to mammals, nor flowering plants before angiosperms. Exactly when flowering plants arose may be tweaked and refined, but the core order of events remains the same.
It doesn't matter. The first life form that lead to plants could have been before all of those.
That flowering plants are actually at least 240 million years old (something I didn't know, thanks for the heads up

) is ultimately only so much detail. The order of events, plant ancestry, etc, remains unchanged. It's not like we've discovered the first lifeform was a solitary rose.
My point is that using fossil evidence one must realize that we can't know how far back certain life forms were present solely on fossil evidence. We might not find a rose currently. If there were no other life appearances during the earliest era of earth that was not destroyed, we can assume that there would not be evidence of a rose or flowering plant in the fossil record. That doesn't mean that the first life form that lead to plants did not exist very early. In fact, there is evidence (still controversial) that may show that plants (very simple and primitive) were on earth prior to the Cambrian Explosion.
'Wrong'? What evidence was wrong, exactly? The 140 million year figure is the oldest known fossil, and is the minimum age that flowering plants can be. But there is a maximum age, too - we won't discover 5 billion year old plants, because there were no plants before there was a planet to sit on.
Ok wrong was the wrong word.
Well, we know. We know the ancestry of modern flowering plants. We know that plants and animals are related by a common ancestor, which must necessarily precede both kingdoms.
And?
Again, your problem is assuming that scientific conclusions are unreliable because they're subject to change. Do you reject the conclusion that the Earth is spherical(ish) because science changes? Of course not.
I am not claiming that it is completely unreliable. But things change and new discoveries are made. When the earth was thought flat was it true? So there is truth in the mix and that is really what science can't tell us. It can provide evidence to support what we hope is true, however, we can see from experience that somethings in science may be supported by known evidence and then a new discovery replaces the old view. That is the essence of science. It can and does change to accommodate new information.