Unemployment Insurance is only temporary. You need to find new employment ASAP, both for your own good and for you to qualify for the mortgage loan that you want. Lenders look for continuous emplyment as part of the underwriting process. So, it would be good to find new employment before your employment runs out.
No... you need employment. Once you have employment and if it is in the same field then perhaps. Unemployment benefits run out so it is unstable to give a loan. After you find a job... go a lender and see where you stand. They will tell you what you need to do to qualify.