You have to rely on Chelsea's board having a change of heart about how the club is run.
Previously a manager has been sacked for doing very little wrong, now you've spent an absolute fortune in one window, essentially a new first team squad, you really think that Lamps job won't be under threat if he's sitting in 6th position in February?
Context won't matter then! But it's yet to happen, and I can't see it happening, as I reckon only Werner and Chilwell will be starters consistently out of the new signings. Havertz will rotate with Mount, Ziyech with pulisic.
I genuinely don't think so. Chelsea have been trying to find a long-term manager for years now - definitely at minimum at least since Jose came back. Obviously he won the league in year 2 but then he brought shame / a lawsuit on the club, forced our best player to play through injury and alienated him, and had us in the lower half of the table in year 3 so he had to go. Conte came in and hit the ground running, then made things completely untenable for the board - he was insisting on absurd and inefficient signings like Sandro for 75m and Nainggolan for 40m+, then he threw a strop when we wouldn't throw good money after bad. In the end, the working relationship became totally untenable and he was sacked justifiably. That said, the way we went after his compensation package was shameful; we should have just paid what he was owed to have a clean break for all parties. Subsequently, Sarri wasn't sacked, he asked to leave and Juve paid us 5m to release him from his contract.
The last manager we fired scandalously was Ancelotti in my opinion. Every move since then has been justifiable.
So ultimately to answer your question, if we're in 6th I'm not worried about Lampard's job. If we're in 16th, then I would be. The expectation this year is a CL place more comfortably than last year; given that it came down to the last matchday last year there's some leeway there I'd imagine.
Also, Havertz is going to be a guaranteed starter. Ziyech and Pulisic play on the opposite wings so they're not actually in competition. All those we've signed can expect to play regularly (except Malang Sarr, who looks to be going on loan).