A timing antenna is a specific kind of GPS antenna that picks up satellites from higher angles better and rejects reflections and low angle signals. I have one, it works well but its not the antenna I am using on my GPSDO right now. (And my GPSDO works fine with a non-timing antenna).
My timing antenna is made by Maxrad, its the same one that is sold under the Lucent name. Its a "GPS-TMG26N" Internally its a quadrifilar helix antenna. It actually works best without a ground plane. just stuck in the air. You can mount it on a metal or PVC pole, but if its high up you likely want to put a good ground on it and also a lightning arrester. Which is kind of a pain in the neck.
How old is your GPSDO? Newer GPSs would probably do fine with an (active) ceramic patch antenna mounted on a saucer-sized piece of metal (say twice the size of the the antenna itself) on a windowsill or put on a piece of metal sheeting in your loft or like Gyro does on top of a bookshelf or similar on an upper floor. (or even not, my home network's NTP server's antenna is on the ground floor on the table my little used TV set sits on, it works fine. That GPS antenna is passive and it uses a CD/DVD as its ground plane).
Unless you are also planning on also using it for geodesy or similar, pretty much any flat conductive object should do as a ground plane. Size is not critical but having it there to make the antenna resonate at the right frequency is. So, a ceramic patch GPS antenna doesn't just work best with a ground plane - they often perform quite inadequately without one.
You should only need one amplifier. Most GPS antennas contain low noise amplifiers which are powered via their coax cable, so its likely you can't add another one unless you figure out some way of passing additional power through it to the antenna. (could be done with a bias-tee)
But likely you wont need it.
Just try different locations and pick the best one.