Yes, they might detain asylum seekers at the port of entry in the US if they don't have a valid visa or a parole or if they are inadmissible. This depends on the individual case.
They will remain inadmissible even if they are granted asylum. The grant of asylum doesn't automatically waive their prior immigration violation (illegal entry).
They will be subject to the inadmissibility bar under section 212(a)(6)(A)(i)
They will have to seek a waiver when they apply for adjustment of status (green card). A waiver might be granted for asylees and refugees for humanitarian reasons, family unity, or a public interest.
Under the current USCIS policy, form I-602 is not required to waive the inadmissibility bar section 212(a)(6)(A)(i) and 212(a)(9)(b) unless the I-485 adjudicator asks for it.