Electronic – What are the drawbacks of transmitting RF frequencies through an antenna that is designed for different frequency

Yes, you will need an external RF source of some sort (ideally a sweeper) and a directional detector (e.g. http://www.ebay.com/itm/HP-Agilent-85021C-Directional-Bridge-10MHz-18GHz-/301374539011) or directional coupler and detector (e.g. http://www.ebay.com/itm/HP-11664A-18Ghz-Detector-for-HP-8756-HP-8757-/380853230821). These units have no RF electronics in them at all; they are basically glorified power meters - you need to supply the source and the detectors. I have not used one of these myself, but I think the way it's intended to be used is you put a splitter on the source so part of the signal is sent to a detector for the R (reference) port, then you can put a directional coupler and another detector between the other port on the splitter and your DUT port to measure the reflected power. Other detectors can then be used for transmitted power out of other DUT ports.

Some of the older VNAs need an external source as well, but it gets sent in through the back of the test set instead of the front.

Antennae are mostly symmetric in their receive and transmit behaviour, i.e. they will have the same gain and frequency response in each role.

The main place where the symmetry breaks down is power handling: an antenna designed for receiving needs to carry only a tiny quantity of power whereas a transmitting antenna, e.g. for a large radar or a long-range communication link, might be carrying kilowatts or even megawatts of instantaneous power.

Power is not an issue for garage door opener radios though ;) However, I would not necessarily expect that replacing the existing wire-antennae with these new antennas will yield any improvement. They may do, or they may be worse depending on how well their impedance is matched to radio circuits you're using them with; have a google for VSWR. The relationship between the antenna you use and any ground plane below is probably more important than changing one cheap piece of wire for another slightly-less-cheap bit.

By all means experiment with those, but don't have high hopes for longer range.