The problem you have with single symbol transmission is clock synchronization.
This is a problem for any transmission standard.
In a slow AM, FM or QAM transmission you might get away with slightly desynchronized clocks as long as you sample at any point where the symbol is valid on the receiver side (which is basically the symbol time minus the transition time caused by the frequency response of the path between sender and transmitter).
With OFDM, you have an additional clock that needs to be synchronized: the modulation clock for the subcarriers. If that one is off, you end up with a phase shift, i.e. a rotation of the demodulated signal, which will make the signal unrecoverable for any modulation scheme that looks at "absolute" phase, like QAM or φM.
So, OFDM always includes clock recovery mechanisms, like a pilot tone or pilot symbols, that allow reconstruction of both master carrier and subcarrier clock, both of which will drift between sender and receiver based on external influences like temperature, and need to be constantly adjusted.
In a continuous-transmission scenario like DSL, clock recovery is designed with minimal overhead and a longer training phase (which is why it takes so long for a DSL line to come up after turning on the modem). Packet transmission like WiFi, where there are multiple stations with wildly diverging clocks on the other hand optimize for a short training phase (at the beginning of a packet), which requires each packet to begin with a fixed preamble that doesn't contain meaningful data.
LTE is a bit in between these two, because it separates channel allocation from transmission standard, so the actual transmission doesn't need to split into multiple subchannels (as long as sender and receiver agree). The channel allocation is still expressed in multiples of the subcarrier clock, and you need to be synchronized enough to negotiate your channel allocation, so that is pretty much a moot point.
In the real world, all of that is pretty much academic, because there isn't much of a use case for single symbols. Packet switched networks have minimum packet sizes because packets need to contain addressing information, and that information needs to be repeated for each single packet. Leased lines transmit idle and pilot symbols because it allows the clocks to remain synchronized, so the next payload packet can be transmitted without a training phase, which would introduce a delay.
To answer your questions:
You can transmit a single OFDM symbol, which consists of one symbol in each subcarrier. You will have to sacrifice some subcarriers for synchronization information to allow the receiver to recover the clock.
802.11 minimum frame length is given by preamble + addressing information + checksum. Data transmissions are pretty much optional, but frames without data are not usually sent because AFAIK there is no upper layer protocol that assigns meaning to an empty frame (the protocol number inside the Ethernet frame would tell us which upper layer protocol would handle the empty frame though).
LTE minimum frame length is one slot because there is no point in doing shorter transmissions, the slot is used anyway, and if the packet doesn't use it fully, it is easier to just fill the rest with a fixed value rather than think about how to pass the "invalid symbol, do not generate subcarrier" information all through the stack.
LTE can in theory generate single-symbol transmissions on subcarriers because the clock recovery is handled on the encapsulating protocol, but that is equivalent to a WiFi frame that contains a full preamble, address, checksum and a single byte of payload data: what is one symbol on one layer may require multiple symbols on another.
There is some leeway in OFDM signal generation that is used to optimize the generated signal's crest factor, but that doesn't happen on a per-symbol basis because it is too computationally expensive for that.
