Lets start off by using the right terms, the gadget thats in the car is the transmitter, it transmits a coded signal that the RECEIVER in the garage picks up to operate the door.
It would be best to connect your coax centre conductor in place of the existing aerial wire. Your current aerial is likely to be too short so the receiver is being fed from a small capacitor, using the new aerial assembley, it would be fed from 75 ohms, which might not be a good match, but as you will be getting , say, five times more signal, it will not matter if you lloose a bit.
Yes the aerial forms a T, but it is imperative that the wire bit is in two parts, else you will have a short across the coax and hence across the aerial input to your receiver.
As for support, I would use drawing pins ("thumb tacks?") and tack the aerial along the horizontal wooden frame above the door. If the ceiling joists go in the correct direction these would do, else string it from bottom of joist to bottom of joist.
The most difficult bit is attaching the coax to the PCB, I found some miniature RF coax and used that with a BNC to BNC adapter to get to the standard 6mm TV coax used for the aerial. What I actually did was to strip the PVC from the coax for the .25m, then fiddle the centre conductor and insulator out of the side of the copper braid "tube" and drawing pin the ends of the aerial bit to my shed ceiling, with extra pins to support the coax.
Frank