Q1.1: Why cannot use the helix calculators for rubber ducky antenna?
Because the helix calculators have different goals. A helix antenna as understood by these calculators is meant to be a directive antenna, whereas the average rubber ducky antenna is just meant to be a short monopole.
In short: because one is meant for designing directive antennas, the other is not.
Read the wikipedia article closely! Normal-mode vs Axial-mode.
Q1.2: Even as rubber ducky antenna for a certain frequency, there are various dimensions available, with different electrical length. Then, how to choose?
Based on datasheet data matching your requirement. In antenna construction, there's more than one way to achieve something, and you always have to do trade-offs between things like bandwidth, efficiency, mechanical stability, weight, …
Oh, and cost. If you're buying antennas, you might be interested in the cost :)
Q2.1: In real applications, the ground plane (the PCB to mount the antenna) would probably be smaller than the reference ground plane from the antenna vendor's design
um, depends on the vendor's reference design. There's a lot of vendors of handset-related components (for example, SiLabs sells FM modem ICs and they have plenty of appnotes). I'm pretty sure there's designs with "small" ground plane to.
Honestly, if you sell an antenna whose prime application is handset devices, and assume your customers will be carrying around 1 m² of ground plane, you're doing something wrong in terms of RF modelling.
For example, the above mentioned antenna is claimed to have 50 Ohm impedance but they also claim that "Electrical specifications and plots measured on a 7.62 x 19.05 cm (3.00" x 7.50") reference ground plane"
That's not a reference design, that's a measurement setup – and it makes sense to measure with a good ground. Smaller ground, as well as about anything about the placement of the antenna, will lead to mismatch. But that's how it is – if you're building integrated RF devices, antenna matching has to be part of your design process. It's important that the antenna starts out with a usable impedance somewhere close to the (typically 50Ω) impedance that you'd want to use. But matching networks or lumped elements are usually inevitable.
Therefore, does it imply that the return loss, VSWR, and impedance at 433 MHz would be poorer than they claimed?
You use a poorer ground than was used for measurement, you get poorer performance. I don't think that's overly surprising! Monopole antenna manufacturer can't do anything about your ground plane, but give you a design starting point.
Q2.2: Considering the above mentioned situation, the antenna would not match the transceiver modules which claimed to have matched to 50 Ohm. Then, again, how to choose rubber ducky antenna? Will the matching ever be met?
Since rubber ducky antennas are only parts of transmitter systems, yes, you are the one in charge of making a good match. But: you'll have to do that anyway. Not-so-dirty, not-so-secret: Your average RFIC doesn't have perfect 50Ω source impedance, either, so matching is done on that end, too.
So, the reasonable engineering approach here is to have a matching network between the monopole and the transmission line, and one between the transmission line and the RFIC. If things are nice and you need to optimize for cost, simplify or omit one of these two networks by matching the transmission line to antenna or RFIC (instead of say 50Ω); but usually, you'd win little by that and then can't optimize the matching networks individually anymore, so you don't.
Q3: In some cases, rubber ducky antenna is sold together with the radio transceiver module, such as HC-12 (link as below).
I say this every 18.34h, so I will say this again: If you want electronic components, buy them somewhere you get a good datasheet. Banggood, aliexpress, ebay, amazon are not places to buy components. If you've never seen a good datasheet: go to any reputable electronics distributor website and get the datasheet for any name-brand inductor. A stupid, 10ct, SMD inductor. A passive device. If your complex digital transceiver IC comes with less documentation than an inductor, don't buy it.
Here, it would've been especially simple: the module is essentially just the "Figure 1: Application Schematic" from the Si4463 RFIC data sheet. If a supplier can't even link to the data sheet of their central component: Buy somewhere else. They don't have technical staff, and they're distributing their "support load" to "the internet" hoping that strangers will help their customers.
Someone said that there are special matching network on the transceiver module so that the 1 km system range could be possible.
Come on, you know that range isn't only a function of a matching network, but of path loss, antenna efficiency and directivity, intereference level, noise level in the receiver (which is dominated by temperature), data modulation robustness, forward error correction, multipath propagation in the channel, …
So, "someone said" is about as good "as my imaginary friend suggested". Only that your imaginary friend probably is not interested in boasting or selling his products...
But yeah, read that datasheet that I linked to above, especially Fig.1. There's a proper matching network right there in the data sheet. Of course, that point is totally moot if you bring a 433 MHz transmission line to a rectangular through-hole, which simply is not an RF-capable interface at all, and will have a random impedance, relative to the driving transmission line, and can't be matched away.
Since the board design is of unknown quality, and we don't know what it can do due to lacking documentation, no-one can ever tell whether 1km is possible.
I could throw around sensitivity numbers from the datasheet, calculate the free-space loss and antenna gains, but really, due to the black-box module design, we might simply see orders of magnitudes of behaviour differing from a theoretical calculation. So, with that level of documentation, I won't be wasting my time on that.
If I change the rubber ducky for another antenna that resonantes at 433 MHz (since there are both antenna pin and ipex interface on that tcvr module), would the new combination really work?
Nobody can know without trying, because that module comes without any documentation or even a specification of the impedance of the coax connector at all.
So, honestly. I don't even know what this has to do with amateur radio, but don't buy this underspecified stuff, which probably ships with the antenna unsoldered to avoid the need for certification for US/EU imports, which it'll almost inevitably can't get.
Instead, go to the central RFIC's product page. You'll notice there's a lot of documentation there.
You're looking for "AN643: Matching…". That'll tell you how to match your receiving end to the 50Ω transmission line. You'll do the same for the antenna, which has an impedance dependent on ground plane, and placement of other non-1-$\varepsilon_r$ materials around the device (e.g hands and heads on a handset).
You'll notice that a perfect match for a single antenna impedance is nice and all, but you'll need robustness, i.e. a match that is still "OK" when the antenna impedance isn't as expected. So, design your matching network to
