Am eligible to apply for ILR in 2 more months and It'll be through my next employer

[needHelpPls]

Hello,

I'm being made redundant by end of this month though I'm eligible to apply for ILR in Jan 2025, my current visa is valid until mid Feb 2025.So I'm in the next visa sponsored job hunting.I'm exploring both Contract roles & permanent roles.

I like to know the timeframe involved as I have only 2 months left in my current visa.

1.The next employer when they apply for CoS for my employment, is there a sponsorship for short period versus longer period option available for them and is the price different? So as the charge for short period could be lesser, I can let know this info (to my potential next employers)as it's benefit for them to hire me, as they don't need to sponsor again and I can apply for ILR in a month or two pls?
2. What’s the exact timeline for my (next visa sponsoring) employer to apply for CoS to get it? I like to know this timeline as I've limited time and if they don't have a CoS already. This leads to the next question:
3. Can’t my (next visa sponsoring) employer have allocated quota and can have a CoS already to attach for my skilled work visa application? The one of the small company CEO/director says it’s not possible these days and for each candidate a fresh application of CoS to be made to the Home office by the employer. Is it true?

Kindly shed some light on the timelines, as it'd clarify me alot!

Thank you.