Can L1 holder move to H1? Does companies file them these days?

Currently We (My husband, Me and my infant kid) staying in US, I am on H1, kid on H14 and husband is on L1 from his employer. Now the situation of his employer doesnt look promising and they dont have many projects left here to work on. So we are afraid that his company may ask him to return to India.
My situation is little stable, I am H1 and my employer would be filing H1 extension as well for me. Now we have question like Can my husband easily switch from L1 to H1 by switching the employer? (First he needs to search such employer who can file his H1). His profile is high profile and in demand. Do really companies here in US do that in currency economy?
Another question is Can somone work on H4, I believ No but just wanted to ask if there is any new rules and regulations.
Please help with Expert suggestions. We just came to US before 3 months, dont want to go so soon back to India for obvious reason.

Thanks in Advance

I think your husband cannot get H1B this year as the cap has reached on 11th June - he will have to wait until April 2013 for FY2014 quota. However, he could file cap-exempt petition if he finds an employer who is eligible to file cap-exempt petitions.

No person can work on H4 status, legally.

Thanks Sujith, Do you know any good sites where he can search for openings with such employer?

I do not know about any sites that he can use to look for cap-exempt employers.