Whenever samples have been sitting for more than a few hours or have been freeze-thawed, you should always mix them before using them. If your DNA is in microcentrifuge tubes and they've been in the fridge for a week or more, have a look around the lid and sides of the tubes and you'll probably see water droplets (condensation) - if you open them, this will go everywhere.
It's safe to vortex some DNA samples, depending on what they are. Isolated PCR products or cDNA samples can generally be vortexed without damaging them. You should not vortex plasmids (especially big ones) or genomic DNA, because vortexing will shear long pieces of DNA into smaller fragments. You can mix them more gently by flipping the tubes upside down and flicking them. Alternatively, spin down the tubes (to collect the condensation at the bottom) and then mix the samples by gentle pipetting.
No worries! You don't need to worry about gDNA shearing too much if you have plenty of sample and your experiment isn't quantitative - something like amplifying a gene for cloning, genotyping, copy# determination etc - these experiments will all usually work from moderately or even very sheared gDNA, as does library preparation for short read sequencing (Illumina). If you're working with low input samples, or long-read applications (nanopore or pacbio sequencing, in particular) then you'll want to avoid vortexing, mix gently and pipette slowly. Hope this helps!
if its high molecular weight you will shear it by whatever you do. If you are only using it for PCR, vortex to hearts content. If its really goopy pipetting can be a challenge anyway.