The mesh bag will contain the hops and make racking easier. The down side is that the bag may be hard to remove from a carboy. The weight will ensure all the hops are in contact with the beer.
I just put the hops in the carboy, before or after racking the beer. The whole cone hops will tend to float on top. You can swirl the carboy and get those all wetted. Some will sink, some won't, in my experience.
To rack with the free floating hops, I have a big nylon mesh bag that gets boiled, then teh racking cane is inserted, and those are stiffed in the carboy. The nyon bag acts as a filter, keeping the hops out.
Keep the dry hoped beer in the carboy at the 60-70 F range.
There is risk in racking as pointed out. If you have good sanitation techniques, and learn to rack gently without splashing, you can make good beer without infections or oxidation. With a CO2 system you can rack and transfer under CO2, no worries about O2. That takes more equipment and time. I have an Union Jack clone that will be racked to a second carboy today under CO2, just because the first dry hop charge was big, and there is a second dry hop charge that is just about as big. It turns into a volume problem after a while.