How to get up and running immediately from a Time Machine backup and a borrowed computer
You can verify that your data is there by opening the backup drive, then go to /Backups.backupdb/YourComputersName/Latest/. This is a copy of the most recent backup of your disk. You should be able to view your home directory, or mount it in case you use File Vault.
Restoring is as easy as selectively copying files from your backup drive to your or your friend's machine. Also see here.
As for full system restore, this will require a partition on which to restore to. Which is major surgery, so to speak. You definitely want an unused partition for this. Since your friend probably only has a single partition on his Mac, this is not going to be easy. Partition resize utilities are available, but I've had a major commercial utility seen failing and destroying data on a Windows machine, so I cannot recommend that.
You can restore during Mac OS X Setup, and even just selectively e.g. restore Applications and User Profiles, if bad 3rd party drivers were acting up, for example.
Added information after third substantial edit of original question:
Only if your external Time Machine backup drive has a second partition on which you can install your system. Hope that your friend has his OS X system disk available, or prepare for that.
Get an external USB disk and restore to that instead. That will leave the internal harddisk untouched.
Answering the "steps 3 and 4" part, as others have covered the general principles:
Depending on how much you back up (and/or want to restore), there are several options:
To do a whole-system restore, hook up all the relevant drives and then boot from the install DVD. Once you get past the first screen, it'll have a Utilities menu with (among other things) Disk Utility (which'll let you create/erase a partition to restore to) and Restore System from Backup (self-explanatory). Do not proceed with the OS installation, it will have been recovered from the backup.
To do a more selective restore (or if you have the OS files excluded from the backup), install a clean OS from DVD, then when the Setup Assistant gets to the screen asking "Do You Already Own a Mac?", there's an option to transfer from a Time Machine backup -- you can use this to restore user accounts, applications, as well as some settings and misc documents. Note that you can also run the Migration Assistant after initial setup (it's in the Utilities folder) to do restores like this after initial setup.
They aren't as relevant to your question, but for completeness I'll mention a couple more restore methods:
To restore specific files/folders, use the Time Machine icon in the Dock to enter the cool spacey view-through-time mode; find the item(s) you want, press Restore, and they'll pop back into existence. Note that a few applications (I think mainly Apple's Mail, iCal, and Address Book) support this mode to restore their items without needing to know where they're stored in the file system.
Finally, you can open the Backups.backupdb folder and root around until you find what you need.