Do the partitions in the Recommended Partitioning Scheme have to be primary or can they all be logical partitions within the same extended partition?

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The answer to all your questions here seems to be 'no difference whatsoever'. There are operational differences if you want to resize them, but in use there really is no practical difference unless you're trying to do raid or fault tolerant partitions.

The only reason extended partitions exist is to work around the four-primary-partition limit imposed by IDE drives.

If you're looking to learn about how these work, you probably want to look into logical volumes - I find them slightly more tricky to set up in the first place, but far easier to chop, change resize and replace later on.

I am installing Fedora onto my laptop, which already has three primary partitions. I would prefer not to delete any of the partitions (Recovery Partition, System Reserved, and C Drive). If it helps, the laptop is a VAIO F2 series.

This is due to the limitations of the MBR. If you had a GPT (GUID Partition Table) you wouldn't have any problem.

Is it possible to use logical partitions instead of primary partitions for the swap, /boot, /, and /home partitions?

Yes it is. You will not have any problem. You have to create an extended partition (which is a primary special partition/placeholder) and inside this extended partition you can create a large number of logicals. Linux can boot from logical partition, in contrast with Windows which cannot.

If it is possible, are there any significant differences from a learning perspective of using logical partitions rather than primary partitions?

Not really. Logical partitions are much like primaries, but you will notice some differences. First, they have numbers over 5 (i.e. /dev/sda6 , /dev/sda7...etc) , second, they are contiguous. You can have primary partitions like 1,3,4 (here we see a gap/lack of number 2) but you cannot have logical partitions like 5,7,8. If 7 exists, there must be a 6 and a 5.

If it is possible, are there any significant difference from a technical perspective of using logical partitions rather than primary partitions?

Personally, I didn't notice any significant difference, neither a little one. Some people say that the drawback of using a logical partition instead of primary is the low performance, but I really didn't notice something like that ever. I'm using logical partitions as well because of the triple (quad sometimes) boot I have. Though, because the /boot partition would be away of the MBR you will have a few milliseconds delay on boot process.

Any other advice would also be appreciated.

Use the ext4 filesystem instead of anything else (for now). Neither Btrfs or an older ext3 or anything else. If you decide to go with LVM , read,read,read,read a lot before you continue. LVM is very flexible, but personally I find it very complex. LVM it is good if you don't know the space you want to allocate when you create large number of separate partitions (/boot/ , /var/, /root/, /home/ ...) and you can resize them later, on the fly, without any problem.