I am going to disagree with DVK's answer. While it is correct that a wand is not necessary to cast spells, not any object can be used to channel magic. The object being used must have magical properties of its own.
First, confirmation wands are not needed for spell casting (from the Uagadou writing by JK Rowling on Pottermore):
The wand is a European invention, and while African witches and wizards have adopted it as a useful tool in the last century, many spells are cast simply by pointing the finger or through hand gestures.
Second, an overview of how wand materials were selected and that they needed to be a magical substance (from the Mr. Ollivander writing by JK Rowling on Pottermore):
Prior to Mr Ollivander's proprietorship of the family business, wizards used a wide variety of wand cores. A customer would often present the wandmaker with a magical substance to which they were attached, or had inherited, or by which their family swore (hinted at by the core of Fleur Delacour's wand). Mr Ollivander, however, was a purist who insisted that the best wands would never be produced merely by encasing the whiskers of a favourite Kneazle (or the stalk of a Dittany plant that once saved a wizard's father from poisoning, or the mane of a kelpie a witch had once met on holiday in Scotland) in the customer's favourite wood. The best wands, he believed, had cores of immensely powerful magical substances, which were expertly enclosed in specially selected and complementary wandwoods, the result to be matched to an owner with whom the wand itself felt the most affinity.
We also see that the wood itself could not be from any ordinary tree either (from the Wand Wood writing by JK Rowling on Pottermore):
Only a minority of trees can produce wand quality wood (just as a minority of humans can produce magic). It takes years of experience to tell which ones have the gift, although the job is made easier if Bowtruckles are found nesting in the leaves, as they never inhabit mundane trees.
IIRC, the only non-wand object we see in the books is Hagrid's umbrella, as you mentioned. However, he is only using the umbrella to hide the broken wand. The umbrella itself is not being used to focus the magic.
You also asked about the Sword of Gryffindor. While the sword was made from Goblin Silver, which does have magical properties, there would be no way for a core to enclosed inside it. Thus it would not be able to focus magical energy.