Hiring a non-EU is more work and cost for an employer, so if they're not accustomed to doing it they might be reluctant. Medea will be along to link to what an employer has to prove in order to hire you. AFAIK employers don't get to apply specifically for B or L. They apply and the authorities decide what you get. L permit is common if it's a short-term (i.e. 1 year or less) contract.
Permit processing time varies wildly, especially with the Corona situation thrown in. 8-12 weeks is common, or was before all this.