I ran a reaction using THF as solvent. Can I use ethyl acetate for extraction to get my product out?
yes. You'll probably quench the reaction by adding some aqueous acid or base or some such thing. This will create a biphasic mixture: crude material in THF, water-soluble byproducts in water. You'll separate them in the sep funnel, and put the aqueous layer back in the sep funnel.
Then you'll extract the aqueous layer with EtOAc. You're extracting trace product out of the aqueous layer using an organic solvent, and EtOAc is usually just fine. The only time it might be a problem is if you're quenching the reaction with strong acid or strong base. You're likely to hydrolyze the ester and produce acetic acid - which may be detrimental to your system, and at the least has a much higher boiling point than EtOAc.
Yes, oils of all viscosities are quite common in organic synthesis. Don't worry too much about making it a solid. Just note that it's a viscous oil and move on.