David,
I answer boldly "yes" to your first two questions, but read at the bottom for my take on the third one.
In my study of irregular or guerrilla warfare in Civil War Missouri, the big push was on for 1864 to benefit from Confederate General Price's long-awaited, and widely publicized long beforehand raid of Missouri in fall of 1864. Part of the big push was the sending behind Union lines starting as early as the winter of 1863-1864 scores of Confederate recruiters and their cadre to round up and sign up as many southern men as they could.

--In some areas, such as the Monroe County area, Major Elliott Majors of Colonel Caleb Perkins' command (I think that's right) signed up scores of men in the summer of 1864 and had them remain at home until he sent word for them to rendezvous later.
--Further east along the counties lining the Mississippi River, master recruiter Colonel Caleb Dorsey (who arrived in May with over 100 veteran cadre after a mostly uneventful ride up through southwest Missouri) gathered scores of men with his stealthy, quiet approach mostly in Lincoln, Pike, Warren, St. Charles, and Montgomery Counties, but he took them to Arkansas after Price's army passed by. Colonel Dorsey set out using the unseasonable, severe snowstorm of 3-4 November 1864 to help cover his crossing south of the Missouri River and avoided Union bases and patrols, keeping to ridgelines all the way to Arkansas while Union forces were still returning to their assigned bases after the pursuit of Price's main force.
--Another late-comer was Colonel Eli Hodge and his recruits from around Boone County who also set out after Price's defeat and sadly, were set upon near the Arkansas border by two or three regiments of Union cavalry and suffered heavy losses. I think Hodge's neophytes were part of Colonel Caleb Perkins' overall command.
--Other successful recruiters in the Clay and Ray County area were Colonels Lewis C. Bohannon and Boaz Roberts, and I think they were able to bring their recruits to Price's army as it passed by south of the Missouri River.
--One of the least successful recruit teams was that of Colonel John Winston and LTC John Calhoun "Coon" Thornton who arrived in the Platte and Buchanan County area in the winter of 63-64, but Union troops captured Colonel Winston at his home on 21 March 1864. LTC Thornton carried on, but estimated Price's invasion too early and triggered an insurgency of at least 600 recruits on 7 July, many of whom were turncoat members of the questionable "Paw Paw Militia" of the 81st and 82nd Enrolled Missouri Militia (EMM). The Union leadership sent most of four cavalry regiments and lots of local EMM who swept eight counties north of the Missouri River during the rest of July and killed or captured scores of Thornton's men. The remnant in late July and early August quietly passed south of the Missouri River in small groups led by veteran cadre and made it to Confederate forces in Arkansas a few weeks before Price launched his invasion in September 1864.
--Another less successful recruiter venture was that of Colonel William F. Peery and his small cadre team of three on their way to Livinston County. On 29 September 1864 elements of the Union 65th EMM captured and evidently murdered Colonel Peery and two of his subordinate officers in north Carroll County, and the EMM wounded but captured alive the fourth team member. The miliitamen who killed these Confederate officers were probably carried away by hysteria about Price's raid which had been in southeastern Missouri since 19 September, but that is hardly an excuse to kill these officers while they were evidently attempting to surrender.
--There were numerous other recruiting teams across southeast, northeast, and northwest Missouri that year. The population of southwest Missouri was noticeably reduced by 1864, so only a few Confederate recruiters operated there.

In short, yes, literally hundreds of recruits rode to join Price's army as it passed through Missouri or rode south to Arkansas in the days after Price's army passed by. As indicated above, Union forces killed or captured hundreds of other recruits attempting to reach Price.
Yes, many of those who successfully made it to Arkansas after Price's raid remained in Confederate service until the mass surrenders in May and June 1865 across the Trans-Mississippi Department.
To answer your third question, I would imagine the names of those new recruits who joined old or new southern units would be reflected in the written record. Bear in mind that the Confederate records in the Missouri State Archives are full of holes despite the efforts of the Missouri Adjutant General's Office a few years ago and more recently in the Missouri Department of State to attempt to enter the overlooked names into the record.
Bruce Nichols