I am working in the information security industry and personally following medical technology closely. Your first question kind of asks to predict the future, which nobody can. Yet, I will give you my personal opinion on the matter.
Is it reasonable to expect the cost of sequencing an individual's genome to drop so much in the near future as to allow for a new kind of product to be developed and sold in the coming years with which one could perform personal genome sequencing at home?
I would estimate that genome tech and genome research is still growing steadily and more practical uses and (medical) applications are yet to be found. Along with that, current productivity (see this graph), effectiveness and accuracy can always be improved.
That said, there is apparently a big need for this technology and as there is a need, there is inherently a need for innovation of that technology. Innovation often means increase productivity (doing more with less). The Moore's law (although currently controversial) is the observation that the number of transistors in dense integrated circuit doubles about every two years. That said, it applies directly to all technology that needs transistors (which is mostly all technology). If you want another great example of technology where effectiveness has gone up and cost has gone down drastically, check out the development of solar technology in the last 60 years (see this article). So in short, yes it is very likely that the cost will keep going down the coming decades.
What alternative options and safety mechanisms can we implement (or foresee to be implemented) at a personal and collective level to eliminate this privacy violation concern and protect the genome data on public and private servers?
Answering from an information security perspective this question can also only be answered in a very subjective way, meaning it is only an opinion. Data is data and humans are human. Sounds silly but I will explain. Ideally, you would want all sequencing devices to implement solid, future-proof and quantum-safe encryption on the data processed, stored and exported by the machine. Yet, there are two problems here.
First, data is data. Cheap tech means a consensus, when tech makers want to compete they could try to use weaker or no encryption at all. Also, how will this be enforced? By law? Then, who will enforce that law? What if people entirely make the device themself? What if the data is initially encrypted but later in the process handled or stored insecurely?
Secondly, humans are human. As with all data protection and information security, the human is the weakest link in the chain. Data might be handled correctly at first but can you expect from an average user to deal with complicated technical processes and will it still be as secure when those processes are made extremely simple and user-friendly? By law, at least in the European Union with the General Data Protection Regulation (GDPR), companies are responsible to protect your data. But, once they hand it over to you in a secure way and destroy it themself, it becomes entirely your own responsibility.
Additionally, what if (some) people want to share this information for medical reasons. Imagine the massive amounts of data, where a great number of correlations can be found between diseases or where new unidentified diseases and currently unknown causes of diseases can be found. Imagine the information you'll get from the DNA of people with genetic immunity to certain diseases.
I will exaggerate a bit now, my apology for entirely ignoring the privacy aspect here but: Would it be ethical to keep all this data private? Or is the only ethical thing to make it all part of one global scale medical research? Imagine the practical use cases of all this data. Should it be an acceptable risk? What is the actual risk of "leaking" your DNA data anyway? Using global research results for designer DNA with CRISPR? You leave your physical DNA everywhere you go (see this article). Do we clean up all our hairs, skin cells, other bodily fluids that we left in all public places to prevent people from physically finding it? No! So, why would we do otherwise digitally in the information age?