Do the lines automatically start in FAST mode?
Yes.
On FTTC, the line is in a mode that lets it go as fast as it can, without FEC or interleaving.
Unless it sees severe instability, DLM will leave you alone on the day of install, and the day afterwards. If it decides it is going to intervene, it will normally do so on the day after that (ie 2nd day after the install), between 1am and 7am usually.
If DLM intervenes, it might have to tune the setting it chooses on subsequent days.
So it is always good to try to get the IP profile value out of BT's speed tester during the first day or two, before DLM intervention, and again on the days after that.
Example:
On my first install at our old house, the initial install gave full sync speed, but 3-4% packet loss; DLM intervened (adding FEC + interleaving) after close to 48 hours. It dropped speed by about 3Mbps, but fixed the packet loss. This was then repeated a few days later, when my line was regraded from a 2Mbps upstream to 10Mbps upstream, and DLM was reset.