A purported Taliban spokesman, Mohammed Hanif, called AP on Saturday to claim responsibility for both attacks.
In the north, an intruder killed a lawmaker in his home.
After Saturday's roadside bombing in mountainous Kunar province, insurgents opened fire at the troops, a US military statement said.
The wounded soldiers were airlifted for treatment to Bagram, the main US base in Afghanistan, it said. Their condition was not immediately available.
The suicide bombing happened on a main road in Maywand district, Kandahar province, when the convoy of Western and Afghan forces was passing. None of the soldiers in the convoy was believed to have been hurt, said General Rahmatullah Raufi, the local army chief.
Security forces shot and killed an alleged accomplice of the bomber as he was fleeing the scene on a motorbike, said General Rahmatullah Raufi, the local army chief.
In other violence, an unidentified assailant fatally shot dead Sayed Sadeq, the speaker of northern Tahhar province's governing assembly, after breaking into his home on Saturday, said Ghulam Hazarat, the deputy local police chief.
Sadeq was well respected in the mountainous region and was a supporter of the country's US-backed central government.
Before the Taliban were ousted in late 2001, he was a mid-ranking commander in a militia run by the former prime minister Gulbuddin Hekmatyar, who is now wanted as a terrorist by the US.