The Guide to the Green Card http://www.entemp.ie/publications/labour/2007/guidegreencards.pdf says the applicant pays it, and the applicant can be the employee or employer. Since the card is issued to the employee and the employee is free to move between employers, I would say that employers would be reluctant to pay it depending on the business they are in and value of the employee etc. It is the same with new the Work Permit system http://www.entemp.ie/publications/labour/2007/guideworkpermits.pdf. It seems to be change from the origional work permit system, where the employer had to pay, but it was very hard to change employer.
The most common thing I've seen is that the company pays but you have to agree to pay it back if you leave the company within X period (one/two years). Also, if Green Card expenses are greater than mere visa renewal (if that's an option), it's not unfair (IMO) to have the employee pay the difference.
"Should" is a moral term - company certainly does not *have* to pay. It's a matter of negotiation/agreement.
I think it's about 8 weeks - the Dept of Immigration has the lead time on its website (says the filing date of the applications they are dealing with at the moment) so you'll get an exact idea from there: http://www.entemp.ie/labour/workpermits/index.htm