a) Upgrading from MS-DOS to Windows 8 on actual hardware!?
As long as the hardware meets the Windows 8 OS requirements (they are higher than Windows 7 as it uses an x86 extension instruction set for better Kernel performance) you can install Windows 8 on the system. This rules out Pentium 1, 2 and 3 processors as they were before this extension existed but I believe some Pentium 4 processors are compatible.
You do not need to use DOS to install Windows 8. Windows 8 should come with it's own bootable install program that allows you to install it on systems with no bootable operating systems present. I would advise this route rather than trying from within DOS since I do not think the OS discs come with a DOS compatible installer executable on them (only a Windows NT Kernel compatible one).
It should be noted that if you want to parallel install DOS and Windows 8 (some academic or legacy institutes need this as they still run programs only compatible with DOS) you need to install DOS on the C partition and have Windows 8 on another separate partition. It may need some complex tricks (which I do not know of) to allow DOS to co-exist since DOS does not support installing on non C letter partitions.
Do note that the partitions used by DOS should be overwritten in the process since it is not a very high performance partition format compared to the default used by Windows 8. Remember to back up all your data on a USB stick or as part of a hard disk (since DOS does not support large partitions?) before going ahead to avoid data loss.
2) How long it takes to upgrade from MS-DOS to Windows 8?
Seeing how you will not be installing Windows 8 through DOS, it will take as long as it would on a clean system with no OS installed. Maybe slightly more if you can do something complex with the partitions to prevent data loss.
In real time no more than 2 hours max and that would have to be on some very low performance backing storage device.
RUNNING MODERN APPS in Windows 98!? Possible?
Yes it is possible if the application is built to be Windows 98 compatible. Few developers do this however since Windows 98 has no support anymore and supporting it may require programming practices which do not allow for optimum performance.
If the application is written in a language like Java or Python then as long as you can get the execution environment (or build it) for Windows 98 it will work. I do not think Oracle supports Windows 98 anymore as it has well exceeded its end of life but I think the open Java VM from GNU may with the right build flags (will perform badly due to limitations of Windows 98).
I would advise updating the system to Windows 7 or Windows 8. There really is no excuse to run such old operating systems. You may be even better off just putting Linux on the system instead as that may not have the same performance as Windows 7/8 but it will at least perform better than 98 and be more secure/support more modern programs.
2) What is the most destructive malware currently?
Malware written by government institutes targeting computer controlled mechanical devices such as atomic centrifuges or power stations. By incorrectly controlling the involved mechanical machines it is possible to cause hundreds of millions of pounds of damage within a few seconds without half the risk or cost that a conventional military strike has. The fallout from a destroyed atomic reactor has the potential to kill millions of people as the result of over exposure to radioactivity. Other possible applications include taking control of enemy military drones, firing enemy missiles and disabling enemy production/communication, both with huge destructive consequences.
In the private sector the most destructive financially are simple key loggers and phishing malware to mine personal information. Billions of pounds are stolen from the UK alone every year as a result of personal information related fraud.
The time of malware permanently destroying huge numbers of computers (late 1990s to early 2000s) is over as a result of changes in OS design practices and the introduction of hardware based OS security features. New backup practices also have helped minimize the loss of data which can be costly depending on the value of the data. Even if the OS is damaged beyond repair one can always reinstall it (hardware is not designed with self-destruct unlike those silly American drama programs want you to believe).