linux from the ground up and throughout it's unix ancestry is intended and designed for the exact purposes that you describe. tightvnc may not be the right tool for the task that are are describing. simply put, remoting into a linux server is a rather simple matter. google is your friend for this one. now, getting your windows machines (which do require quite a bit of additional work and probably some expensive tools) to talk nicely with your linux servers may be a bit trickier.
rdesktop can handle remoting into an xp pro, win2000, or vista pro machine. it's in the repo, but that version is a lil old and buggy. we use rdesktop 1.6 at work, building a package is rather simple for that particular tool. google vectorlinux sbbuilder for instructions. most of the rest of your windows setup will depend on what your trying to do. if your trying to run windows servers, ask your self: why? windows desktops are more understandable. there are a few corporate tools that require windows. in our environment we have two such tools. qcterm to access our old mpe/ix servers and ms office (our sales contracts have not been ported to open office yet, it's a lawyer thing). for qcterm, wine actually works better than windows. for office, we set up one win 2000 terminal server and use rdesktop to access it company wide. we do have four mpe/ix machines (two production, two backup). the rest are all running linux.
unfortunately, the web developer with the lowest bid is an ms fanboi, so our corporate website as well as our intranet site heavily rely on ms software. those are running on the win 2000 server. unfortunately, the company spent a lot of money on an intranet site that most of the company can't access. hoping google-chrome will fix that, so far so good, just a few bugs left in that package to work out.
the thing is, and the moral of the story is, set your ms machines up the way you normally would. linux is configurable enough to adapt to almost any situation. i don't worry about the security stuff in linux, i setup a cron job to update each linux machine every night. if you want tighter control over what gets patched then setup your own repository server and vet each package before you make it available. but, i tend to figure that most distro's are pretty good at making sure a patch won't break something before releasing it.
as for your specific questions:
- How do you assure that the Linux OS of a server is ok? In other words, how do you define a Linux OS as ok? What are the minimum core services required?
i don't understand this question. if your asking what services need to be initialized during startup, that depends mostly on what your intentions are for the server.
- Is there any tools to backup a Linux OS so that if anything occurs, we can use the backup file for disaster recovery to revert the system to how it was before the incident occurs?
most assuredly, there are various raid options built into the kernel. also is't fairly easy to script nightly or hourly backups. load balancing accross multiple servers (hey, if the software is free and cheap, you can use two or three times as many servers) most of this stuff is in the repository, the rest can be googled.
- Is there any cross-platform tools to remotely access and control both Windows and Linux because we will do technical support remotely. I searched google for a while and found TightVNC, but haven't had a chance to try it myself.
windows terminal server, rdesktop and i believe there is a linux terminal server project (xrdp) that uses the same protocols as the builtin windows remote desktop tool. also, a google search revealed openwbem which might fit the bill.
linux server and desktop administration is not as complex as it's ms conterparts. sure you might have to get into the commandline every now and then. but if your an admin and your scared of the commandline, well, you should be looking for another line of work. like i said before, just set up the windows machines the way you normally would, then configure your linux machines around them. i'll almost guarantee, if your linux machines are setup correctly, they won't be the ones that keep yuo up at night.