No. For one, you have harmful oils on your skin that would irritate and harm your frog. For another, you can very easily break their bones or kill them in your attempts to hold them. For another, they don't like being held, they're wild animals. They're much more fun to watch, anyways. Yes,you can. You just have to be very careful! I have some and it is very hard to hold them because they are most likely to jump out of your hand and start hopping away and if you don't put them back in time they will die. But, you really shouldn't hold them.

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Bettas and African dwarf frogs do well together, but make sure your tank is large enough (at least 2.5 gallons) and there are plenty of hiding spots for the frog. Also, watch them when you feed to make sure that the betta isn't eating the frog's food, and be extra careful that you keep the water clean as frogs breathe through their skin.

(For the record, ADF are not the only thing you can house with bettas. They are also fine with mystery snails and gentle community fish that do not have long colorful tails (ie a female guppy or maybe a molly). Female bettas are good with each other and pretty much any gentle tropical fish.