Why are backyard and backseat single words while front yard and front seat are two? It seems exceedingly strange to me.
How do dictionaries handle 'back' words and 'front' words?
Consider the set of fused nouns listed in Merriam-Webster's Eleventh Collegiate Dictionary (2003) that start with back—words that are rendered as closed-up (nonhyphenated) single words:
backache, backbeat, backbench[er], backbiter, backblock, backboard, backbone, backbreaker, backchat, backcloth, backcountry, backcourt, backcross, backdrop, backfield, backfill, backfire, backfit, backflip, backflow, background[er], backhand, backhoe, backhouse, backlash, backlight, backlist, backlog, backpack[er], backrest, backsaw, backscatter, backseat, backset, backside, backslap[per], backslash, backslide[r], backspace, backspin, backsplash, backstabber, backstay, backstitch, backstop, backstory, backstreet, backstretch, backstroke[r], backswimmer, backswing, backsword, backup, backwash, backwater, backwoods, backwrap, backyard
All of the words in this list are instantly recognizable combinations of back plus another extant word. Now consider the set of fused nouns in the Eleventh Collegiate that start with front and (again) yield closed-up (nonhyphenated) single words:
Frontispiece is a bit of an oddball here because, according to Merriam-Webster, it came into English (via Middle French) from Late Latin frontispicium ("facade") already a single fused word. Nevertheless, it has some characteristics of a fused word pair, and frontcourt looks lonely by itself.
The Eleventh Collegiate also lists a few hyphenated nouns that may eventually lose their hyphen and become fused single words. Here are the ones that begin with back-:
And here is the one that begins with front-:
Finally, some nouns consist of two separate words but have a sufficiently distinctive meaning that the Eleventh Collegiate includes entries for them. These have back as their first element:
back bacon, back burner, back channel, back dive, back judge, back matter, back mutation, back order, back room, back talk
(Note that this list excludes noun terms such as back door, back end, back office, and back stage that the dictionary excludes on grounds that the two words together don't connote anything special that one wouldn't surmise from adding the meanings of the two individual words together.)
And here are the two-word nouns beginning with front that have entries in the Eleventh Collegiate:
front bench, front burner, front dive, front end, front line, front man, front matter, front money, front office, front room
Conspicuously absent from these lists are both front terms that the poster asks about: front seat and front yard. But the overall list of inconsistencies between front words and back words is somewhat larger. Here are the pairs that receive inconsistent treatment in the dictionary:
backbench and front bench; back end [not listed] and front end; backlist and front list [not listed]; back line [not listed] and front line; back office [not listed] and front office; backseat and front seat [not listed]; backyard and front yard [not listed]
There are also undoubtedly many pairs (such as back end and front end) both of whose terms are excluded—but in those cases, we can assume that Merriam-Webster views both terms as being composed of separate words.
The American Heritage Dictionary of the English Language, fifth edition (2011) often agrees with the spelling and entry choices in Merriam-Webster's Eleventh Collegiate Dictionary, but not always. These noun terms appear in the Fifth AHDEL but not in the Eleventh Collegiate:
backchecker, backdoor, back end, backhaul, back nine, back office, backpressure, backpropagation, backrush, backshore, frontlist, front nine, front yard
These noun terms appear in the Eleventh Collegiate but not in the Fifth AHDEL:
back bacon, back burner, backblock, backcloth, backfit, backhouse, back judge, back order, backstreet, backwrap, front burner, front dive
And these noun terms receive different treatment in the two dictionaries:
backchannel [AHDEL] vs. back channel [MW]; backformation or back formation [AHDEL] vs. back-formation [MW]; backroom or back room [AHDEL] vs. back room [MW]; back seat [AHDEL] vs. backseat [MW]; backyard also back yard [AHDEL] vs. backyard [MW]; frontline also front line [AHDEL] vs. front line [MW]; frontman also front man [AHDEL] vs. front man [MW]; frontrunner [AHDEL] vs. front-runner [MW]
Why the inconsistent treatment?
The first thing to observe is that, whereas Merriam-Webster inconsistently endorses backseat and backyard and (by omission) front seat and front yard, American Heritage does not. The Fifth AHDEL mentions back seat, backyard also back yard, and front yard, and (by omission) it endorses front seat. So it is quite possible to follow AHDEL style and use word pairs that are not inconsistent (back seat/front seat and back yard/front yard).
As for the larger question of why inconsistencies such as backbench vs. front bench crop up (in both dictionaries), I suspect that the much larger number of closed-up back words (in comparison to closed-up front words) influences writers' readiness to render other noun terms starting with that word as fused words rather than as hyphenated forms or open two-word terms.
Note that some back words are counterparts not of front words but of fore words (virtually all of which are unhyphenated single words, since fore- is a true prefix)—for example, forechecker/back-checker, foreground/background, and forehand/backhand—and others of which seem to be using back as a short form of backward, as in backflip, backflow, backstroke, and backwash.
Whatever the reason, the number of fused back words is much larger than the number of fused front words, which makes the temptation to close up previously separated two-word terms such as backchannel, backformation, and backroom much stronger than the tendency to close up previously separated two-word terms such as frontman. The latter still happens (as the Fifth AHDEL makes clear) but it happens less often than the former.
Viewed in this larger context, backseat and backyard versus front seat and front yard are part of a pattern of orthography in which back word pairs are far more likely to be fused than front word pairs are.
You can also say "back yard" and "back seat".
When words are said together often enough, there is a tendency for them to merge into compound words.
In English this tends to be a slow, organic process.
Another reason (which doesn't fit your examples) for the creation of compound words is to describe novel concepts, such as new technology. Other languages, such as German, will often create a new compound word from other native words to describe any novel concept. In English, novel concepts are often described using a new single (ie non-compound) word which is actually derived from one or more Greek words. Thus we have for example "helicopter" in English, which comes from the Greek "heliko ptero" meaning "spiral wing". The German word for helicopter is "hubschrauber" which comes from the German words "hub schraube", or "lift screw".
Both approaches have different advantages. In German, there's less to learn - you can translate the compound word back into the simpler words and figure out what is being described. But, it can lead to long, unwieldy words like rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz- "the law for the delegation of monitoring beef labeling." In English, if we wanted to make a single word to describe this we'd probably look to some serial number from the official documentation and call it the CF27 or somesuch, eg "Have you got the CF27 for this beef, mate?"
I'm no English maven, but While I'm not aware of any reference to support it, I've habitually fused words like backseat, backyard, backstreet only when using them as adjectives ("backyard BBQ"), and separated them for the noun form ("in the back yard", with "back" as an adjective; "which yard?" "the back yard"). I suppose this "rule" cannot be applied across the board for the long list of "front" and "back" word discussed here, but I always think about it now when facing such a choice.