yes you can setup the ASA for anyconnect full tunnel so that when the users connect via anyconnect vpn, they send all the traffic including the internet traffic over the vpn to the ASA.
anyconnect Apex and plus are only used with anyconnect 4.x; for anyconnect 3.x anyconnect essentials and premium licenses are used.
please read their release notes to check which client would best suit your requirement.
to be able to use ssl vpn, you need to install anyconnect client on the user pc; if you do not want to use the client you can use clientless ssl vpn which is browser based.