Conditionally merge rows

You could use tidyverse:

library(tidyr)
library(stringr)
library(dplyr)

df %>% 
  filter(X2 != "") %>% 
  mutate(
    extract_name = lead(str_extract(X2, "(?<=[A-Z])[A-Z]+")),
    extract_part = lead(str_extract(X2, "[A-Z](\\s-\\s[A-Z]){0,1}(?=[A-Z]+)")),
    new_X2 = ifelse(!is.na(extract_name), paste(X2, extract_name), as.character(X2)),
    new_X3 = ifelse(X3 != "", as.character(X3), extract_part)
    ) %>% 
  drop_na(X1) %>% 
  select(-extract_name, -extract_part)

which returns

   X1            X2    X3              new_X2 new_X3
1 111  House M. Bab     A        House M. Bab      A
2   2  House M. Cac A - C        House M. Cac  A - C
3 121 Street M. Bak     D       Street M. Bak      D
4 121  House M. Aba        House M. Aba SMITH      A
5 141 Garden Harris       Garden Harris WHITE  A - B
6 141  Villa Thomas       Villa Thomas BURNEY  B - D

Note: I don't think this approach is really stable regarding the regex used. For readability I filtered out some annoying rows containing NA and empty strings, you should remove those parts if necessary.

Here is how we could do it:

Credits to MartinGal for the regex "(?<=[A-Z])[A-Z]+") (upvote!)

1. Replace empty values with NA
2. Use lead to move rows up in X3 conditional on NA else not
3. filter if is not NA in X1
4. Extract the important information with str_extract and regex "(?<=[A-Z])[A-Z]+" -> combine this info with column X2 with str_c and finally coalesce both.
5. Remove the string to keep relevant one with regex and str_remove

library(dyplr)
library(stringr)

df %>% 
    mutate(across(everything(), ~sub("^\\s*$", NA, .)),
           X3= ifelse(is.na(X3), lead(X2), X3)) %>% 
    filter(!is.na(X1)) %>% 
    mutate(X2 = coalesce(str_c(X2," ", str_extract(X3, "(?<=[A-Z])[A-Z]+")), X2),
           X3 = str_remove_all(X3, "(?<=[A-Z])[A-Z]+"))

Output:

   X1                  X2    X3
1 111        House M. Bab     A
2   2        House M. Cac A - C
3 121       Street M. Bak     D
4 121  House M. Aba SMITH     A
5 141 Garden Harris WHITE A - B
6 141 Villa Thomas BURNEY B - D

This is a yucky one:

# Retype the data and nullify empty values;
# use X1 as a key: intermediateResult => data.frame
intermediateResult <- data.frame(
  lapply(
    transform(
      replace(df, df == "", NA_character_),
      X1 = na.omit(X1)[cumsum(!is.na(X1))]
    ),
    as.character
  )
)

# Re-structure the data: 
# interemdiateResult2 => data.frame
intermediateResult2 <- do.call(
  rbind,
  Filter(
    function(y){
      nrow(y) > 0
    },
    Map(
      function(x){
        z <- x[!is.na(x$X2),]
        if(nrow(z) > 1 & is.na(z$X3[1])){
          z$X3[1] <- z$X2[2]
          head(z, 1)
        }else{
          z
        }
      },
      with(
        intermediateResult, 
        split(
          intermediateResult, 
          paste(
            X1,
            cumsum(
              (is.na(X2)
            )
          ),
          sep = " - "
          )
        )
      )
    )
  )
)

# Regex it and hope for the best: 
# result => data.frame
result <- data.frame(
  transform(
    intermediateResult2,
    X2 = paste0(
      X2, 
      ifelse(
        (nchar(X3) == 1 | grepl("^\\w\\s+-\\s+\\w$", X3)),
        "",
        ifelse(
          !(grepl("^\\w\\s+-\\s+\\w", X3)),
          paste0(" ", substr(X3, 2, nchar(X3))),
          paste0(" ", gsub("(^\\w\\s+-\\s+\\w)(.*)", "\\2", X3))
        )
      )
    ),
    X3 = ifelse(
      nchar(X3) == 1 | grepl("^\\w\\s+-\\s+\\w$", X3) , 
      X3,
      ifelse(
        !(grepl("^\\w\\s+-\\s+\\w", X3)),
        substr(X3, 1, 1),
        gsub("(^\\w\\s+-\\s+\\w)(.*)", "\\1", X3)
        )
      )
    ),
  row.names = NULL
)