the memory full may be causing the shut down. Are u able to keep the phone on long enuff to go into ur file manager and see whats taking up space, or delete things, maybe move them to memory card?
The only way u r going to know u have a virus is to download an antivirus and run a full scan thru ur phone, ovi store has free trial apps like kaspersky, f-secure and mobishield. But if u cant keep ur phone on long enuf, a hard reset may be ur only option but it will wipe ur phone clean, deleting everythin.