Wall can be defeated in many ways.
First of all you need to understand that wall isn't made to stop minor incursions but rather large scale invasions. That's because:
Wall isn't complete
On the western portion of the wall, there's a place where one can easily bypass the wall.
That place is called the Gorge. The Gorge is massive, well, gorge. It separates the wall from Bay of ice. There is a bridge spanning above it which is called Bridge of Skulls. Wildling raiders often exploit this weakness in Wall to raid inside Seven Kingdoms.
From ASOS, Jon V
In the west they would descend into the black depths of the Gorge to
make their way around the Shadow Tower.
But as that location is unsuitable for launching a major invasion force and can be defended against small parties by a determined detachment of Black Brothers, the Night's Watch has never been much bothered by it.
Given that nearest Night's Watch Castle is abandoned, The area's defense is responsibility of Shadow Tower whose Commander may call for aid from Castle Black if deemed necessary.
Also this is the only plausible route which could be taken by the direwolves to sneak inside the realm from Northern wilderness.
Night's Watch lacks the men to mount proper patrols
Night's Watch isn't what it used to be. They lack the man to guard the wall properly. Only three castles are manned on the Wall which are:
- Castle Black
- Shadow Tower
- East Watch by the Sea
The other 16 castles are abandoned. Even in its prime days, Night's Watch never had men or resources to man more than 17 castles simultaneously.
From ACOK, Jon V:
"Yes. To be sure." The Old Bear unrolled a map, frowned at it, tossed
it aside, opened another. He was pondering where the hammer would
fall, Jon could see it. The Watch had once manned seventeen castles
along the hundred leagues of the Wall, but they had been abandoned one
by one as the brotherhood dwindled. Only three were now garrisoned, a
fact that Mance Rayder knew as well as they did. "Ser Alliser Thorne
will bring back fresh levies from King's Landing, we can hope. If we
man Greyguard from the Shadow Tower and the Long Barrow from Eastwatch
. . ."
Given this, Night's Watch is unable to effectively patrol the wall and stop small bands of intruders.
They still have rangers however who provide them intelligence about greater threats than few Run-away Wildlings or bands of Raiders.
If Rangers encounter small bands of raiders, they normally deal with them themselves rather than reporting back to their respective castle for further commands.
As Mance Rayder said:
The Wall can stop an army, but not a man alone.
As mentioned already, 16 of the 19 castles are abandoned. All of them have tunnels leading to Wild North. Night's Watch always sealed those tunnels with stone and ice however it appears that some castles have secret entrances which were forgotten when abandoning of the castle took place.
Sam and Gilly used one of those forgotten secret passages to enter the realm from Wildling lands.
From ASOS, Bran IV:
"Why?" Meera demanded. "If there's a gate . . ."
"You won't find it. If you did it wouldn't open. Not for you. It's the
Black Gate." Sam plucked at the faded black wool of his sleeve. "Only
a man of the Night's Watch can open it, he said. A Sworn Brother who
has said his words."
Another way to cross the wall is tunneling. One can tunnel his way into the realm by digging a tunnel through the wall.
Of course a task of such magnitude can't be undertaken by a small band of raiders. You need an army for that to do the labour and protect against the Rangers from Night's Watch.
Arson Iceaxe used this method and while he was digging, he was discovered by Rangers who didn't disturb Arson and simply sealed the Tunnel he had dug from the other side.
From ASOS, Jon III:
Jon knew the tale as well. Arson Iceaxe had been halfway through the
Wall when his tunnel was found by rangers from the Nightfort. They did
not trouble to disturb him at his digging, only sealed the way behind
with ice and stone and snow. Dolorous Edd used to say that if you
pressed your ear flat to the Wall, you could still hear Arson chipping
away with his axe
Gorne's way is a legendary labyrinth of underground tunnels and caves which can be used to bypass the wall entirely and emerge on the other side of the Wall. It is said to have been used by Brothers Gendel and Gorne who were Kings beyond the Wall and lead their people South only to be defeated by might of Starks. It is unclear whether this is myth or fact.
From ASOS, Jon III:
"You know nothing, Jon Snow. It went on and on and on. There are
hundreds o' caves in these hills, and down deep they all connect.
There's even a way under your Wall. Gorne's Way."
Bay of Seals
This is a relatively safer way to bypass the wall if you're not seasick or afraid to get your clothes wet when War Galleys of Eastwatch come to drown you.
Wildlings use this small bay to bypass the wall from Sea and cross on little crude boats. They are often intercepted by Eastwatch or Forces of House Umber.
From ASOS, Jon V:
Along the east coast the raiders most often built boats to slip across
the Bay of Seals.
And of course there's the favorite method of the Wildling raiders. Climbing! It is not technically bypassing the wall but I am just adding it for sake of completion.
Night's Watch has a rule of never allowing trees to grow near the wall as the trees can act as launch pads for Raiders to climb the wall without having to tackle the dangers of the wall itself for greater part of the climb.
Due to shortage of men, Night's Watch appears to have grown lax in this regard. When Jon was returning to realm with Jarl and Magnar of Thenn, they used trees growing next to the Wall to get a safe push in their climb.
From ASOS, Jon IV:
"There they are," Ygritte said, and Jon glanced up to see the first
climber emerge above the treetops. It was Jarl. He had found a
sentinel tree that leaned against the Wall, and led his men up the
trunk to get a quicker start. The wood should never have been allowed
to creep so close. They're three hundred feet up, and they haven't
touched the ice itself yet.
From the same chapter:
The Others take them all, thought Jon, as he watched them scramble up
the steep slope of the ridge and vanish beneath the trees. It would
not be the first time wildlings had scaled the Wall, not even the
hundred and first. The patrols stumbled on climbers two or three times
a year, and rangers sometimes came on the broken corpses of those who
Cases specific to your question
- Will/Gared: Being a veteran member of Night's Watch, he (Will in Show and Gared in books) should have been able to use the magical black gate in Night Fort however it is not known if he knew about the gate. Alternatively he could have just used any of the methods above, depending on tools he had with him and the location he was at. It is not specifically mentioned
- Direwolf: The She-wolf most plausibly crossed the Gorge since it is unlikely for a wolf to climb, swim across sea or tunnel her way in. She must have given birth on the other side of the wall where she was found dead as the cubs were very young at that time, with only Ghost having his eyes open.
- Osha and her band: It is unknown which method they used. It could have been any one of the ways mentioned above.