I haven't worked with plants much lately (and when I do I use a Qiagen kit), but historically *all* DNA work was done on ice or at 4oC. The idea was to "inhibit nucleases", and since enzyme action is, in general, slower with lower temperature, on-ice was the lowest practical temperature to use in a lab. Since then reagents have become more pure, disposables that are certified nuclease-free are easy to get, and current "kit" gDNA protocols are run at ambient temperature, so your protocol may be old and the ice-cold pestle may be unnecessary...you may be doing that for historical reasons only.
As for cold isopropanol, if this is for DNA precipitation, again, historically, precipitation of DNA was done cold. In the 1970s-80s, once the alcohol (1-2 volumes ethanol or 0.6 volumes isopropanol) was added, the mixture was chilled, and the "lore" was "the colder, the longer, the better". -20oC for 2 h to overnight was standard. If you had a -80, even better (10-20 min). If you had dry ice, great, 10-20 min. Back then, it was time for a lunch break once your DNA got to the precipitation step. This was disproven a long time ago (DNA will precipitate within a minute or 2 at ambient temp), and so, again, you may simply have an old protocol.
If you are using a new kit or new protocol, ignore what I just said and ask your question again, providing more detail. If, however, you have an old (>15 years old) protocol, and it works, well, you might keep doing what you're doing.
If you want to change your protocol, please do it systematically (change 1 thing at a time), and always compare it to the old protocol so you can assess your results. That's the likely reason you're still doing it the "old way" - changing to a new way takes a study to be sure it's a better way. Or, of course, you can find a newer protocol and compare it to your current protocol and make sure it's comparable (or better).
Another thing - when you have a protocol, be sure to include the date and the provenance (where this protocol came from and how it was adapted) with it - this will help future scientists know the logic behind the protocols used in a lab.
Using cold Isopropanol ( (-20C) just help during DNA Precipitation. Ice cold isopropanol only increases the rate of precipitation of DNA and helps increase yield of DNA. You can also use ice cold ethanol for DNA precipitation and chilled mortar and pestle protect from Nuclease (DNase) activation !!