There is no such thing as a spent conviction as far as immigration are concerned. The 2013 conviction will show, and it is likely the caution will too. There are three 'levels' on the ACRO report:
NO TRACE = No Criminal Record. NO LIVE TRACE = Old Criminal Record. LIVE TRACE = Recent Criminal Record
If you have either of the latter two, you will need to explain further and likely produce a subject access report. Neither would be fatal to an application if properly declared.