Regarding the brakes:
Your current brakes look like long-reach brake calipers to me (older type which has longer lever arms (less leverage). You have to account for the rim getting closer to the frame when you upgrade to larger wheels (I don't think you'll be able to find 27" wheels new, by far the most common size is 28" (700c aka 622mm inside rim diameter). If you are looking at road bike (or more specifically non-mtb wheels) the size 29" should not appear. 29" is often used for mountain bikes with big wheels (3 sizes in mtb wheels, 26", 27.5" (650B) and 29").
29ers or two-niners are mountain bikes and hybrid bikes that are built to use 700c or 622 mm ISO (inside rim diameter) wheels, commonly called 29" wheels.
"The name "29er" comes from a bicycle called the Two Niner, which was offered by the Fisher bike company in 2001, according to 1998 Mountain Bike Hall of Fame inductee Don Cook."
So 29-er mountain bikes do in fact use 700c or 622mm inside diameter rims. The name is just a bit confusing. Either way as long as you look for "road bike wheels" or "touring bike wheels" or "28 inch wheels" or "700c" or "622c" wheels you should be fine installing them on your bike (assuming there is enough frame clearance to fit the bigger rim/tire. The 28 inch wheels' rims do differ in width , i would suggest going with a similar width to your current rims (the width of the tire should be listed on the sidewall of your old tire and it should give you a good indication of your current rim's width, or you could measure the rim width with a caliper or other measuring tool to get a more exact figure. A wider rim will weight more, be a bit stronger, will allow for wider tires which will result in a cushier ride. If you go too wide the tire might not fit in the frame and it will start rubbing.
You need to account for the change in rim size for determining if you can install a certain brake caliper properly, from 27" to 28" this should be approx an inch in diameter so half an inch in radius.
If you would want to use your old brakes you could probably get away with it on the rear brake (it looks like the pads can be adjusted in upwards direction enough to account for the change in rim size), the front cannot be adjusted much more upward from its current position though..).
For installing the Sora (or any other modern short reach brake):
Check the minimum distance from the mounting bolt of the new brakes to the center of the pads and see if this distance matches the distance from the hole in your frame for mounting the brake to the center of the rim's braking surface.
I'm quite sure it should work (installing SORA) but please check (by measuring or preferably asking your local bike mechanic (which should have the parts so he can easily check compatibility) before buying new brakes.
For more info regarding this please check the part about "Reach" on Sheldon Brown's website: https://www.sheldonbrown.com/calipers.html
You should know older brakes like the ones you have usually use a bolt which protrudes all the way through the mounting hole with a nut on the other end to secure them to the frame, and modern brake calipers of this type use recessed nuts (they recess into the frame). If you want to use the newer style recessed bolts on your old frame you'll most likely have to drill out the holes to accommodate for the size of the recessed nut (the hole will be bigger than the current hole in the frame). On steel frames (which I think you have) this shouldn't be an issue though, just make sure not to drill it too large. For the front brake you could theoretically swap to a different front fork (which is more modern and does accommodate a modern recessed style mounting system (but if you have no other reason to upgrade the fork this might be silly).
Here is a video on how to install recessed style brakes on an old frame:
Regarding the wheels:
Wheels do not come with cassette/freewheel preinstalled in general. You'll have to make sure you get a wheel with the right system (either a freewheel hub if you want to reuse the old freewheel or install another (new) freewheel) or a free-hub body (for installing a cassette (which is the system used on (almost) all modern bikes)).
Free-hub bodies differ in length so you'll have to check if the one that comes with the wheel-set you buy is compatible with the cassette (amount of gears) you'd like to install. This info (compatible cassette speeds) should be listed with the specs/description of the wheel(set).
If you want to keep 5 speeds in the rear you will have to use a cassette (up to 7 gears you usually see freewheels and from 8 and upwards cassettes are used).
If you decide to upgrade to more gears in the back you will need a more narrow chain (which matches the amount of gears in the back (for example a 10 speed chain for 10 speed cassette). You will also have to replace your shifter (the one that operates the rear derailleur), assuming you don't have an index-less aka friction shifter (these can be used for all speeds as long as their range is sufficient (maximum cable movement that can be achieved with the shifter when moving it from max to min (or reverse direction) position).
Many people will say you'll also have to replace your rear derailleur but I've had much success using the same derailleur on many different gear setups. You do have to make sure the derailleur cage is long enough to handle the change in chain length when going from the smallest to biggest gear in the back (some mtb cassettes have a really big range meaning a small smallest gear and really big largest gear).
In summary: (regarding wheels)
You can replace the wheel with a wheel which accommodates either a cassette or freewheel, new wheels will almost always have cassette interface (aka free-hub body).
Usually cassettes start at 8 speed, going all the way up to 12 speed (but up to 11 is much cheaper, 12 is very rare).
When installing a new cassette or freewheel a new chain is recommended (since using old chain on new cassette/freewheel can/will cause faster wear or chain skipping teeth and such issues). Make sure the new chain matches the amount of gears on the cassette/freewheel.
Also upgrade rear derailleur if it can't handle the range (doesn't have enough capacity (measured in number of teeth difference) to accommodate the new freewheel/cassette (if it has a significantly broader range than the original freewheel).
Sometimes a derailleur's limit screws won't allow for enough adjustment to get the same derailleur properly adjusted on a different cassette/freewheel but you can oftentimes simply install longer adjustment bolts to fix this issue (remove old ones, find longer versions with same thread and install).
If you have or would like to use indexed shifters and you change to a different number of gears in the back you'll have to upgrade your shifter to an indexed variant with the correct number of gears (/clicks), you could alternatively install a non-indexed friction type shifter (which should in my experience work with any speed as long as the cable movement range is sufficient (as a fore mentioned))
if you would like to keep using a freewheel (you could for example buy a second hand wheel-set with a freewheel hub) you could upgrade to 7 gears in the back if you would like, this should be as simple as removing the (if installed) old freewheel and screwing on the new (7 speed for example) freewheel. You will need sufficient