West Street subway station was created in 1896.
Penn Station is at 34th Street so it is 20 blocks away from 54th St. You would take the subway two stops north.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street (doesn't matter which you take, the A, C or E). Transfer at West 4th Street to the Downtown F train (of the B-D-F-M, the orange line) to East Broadway.You'll exit the subway at or near the corner of East Broadway and Rutgers Street. Walk one block south on Rutgers Street to Henry Street.
To get to 555 W 57th Street via subway, first determine your starting location. If you're near a subway station, take the C or E train to the 57th Street - 7th Avenue station. From there, it's a short walk west to 555 W 57th Street. Make sure to check the MTA website or an app for real-time service updates and any changes.
Assuming you meant "6th Avenue and 45th Street": the easiest way would be to take the Uptown 1, 2, or 3 trains (the red line) ONE STOP from Penn Station/West 34th Street to Times Square/West 42nd Street. You will exit the subway at around 7th Avenue and 42nd Street. Walk north 3 blocks to West 45th Street. Then turn right, and walk one block east to 6th Avenue.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street.Transfer at West 4th to the Downtown B or D trains (of the B-D-F-M, the orange line) to Grand Street (at Chrystie Street).Then walk 4 blocks west on Grand Street to Mulberry Street: Chrystie Street ---> Bowery ---> Elizabeth Street ---> Mott Street ---> Mulberry Street.
It's not very far. You can just walk there. And you'd have to make 2 transfers to get there by subway, so taking the subway really is not worth it. The address 116 Lexington Avenue is at or near the corner of East 28th Street. Penn Station is at West 33rd to West 34th Street, between 8th and 7th Avenues. The main entrance is at 33rd Street and 7th Avenue. So just walk north to 33rd Street, turn left, and walk west to 7th Avenue.
Walk west to Lexington Avenue and north to 53rd Street, and take the Downtown E train (of the A-C-E, the blue line) from Lexington Avenue-53rd Street to 34th Street-Penn Station.
There is no "121 Houston Street." There is a 121 East Houston Street and a 121 West Houston Street.The closest subway station to 121 East Houston Street (at 2nd Avenue/Chrystie Street) is the F train (of the B-D-F-M, the orange line) to 2nd Avenue-Houston Street.The closest subway stations to 121 West Houston Street are the C or E trains (of the A-C-E, the blue line) to Spring Street (at 6th Avenue) or the 1 train (of the 1-2-3, the red line) to Houston Street (at Varick Street).Coming from either station, you'd walk up to Houston Street and then east to 121 West Houston Street. Spring Street is a couple blocks south of Houston, and the Houston Street Station on the 1 train is actually on Varick Street between King and Houston Streets.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street.Transfer at West 4th Street to the Downtown F train (of the B-D-F-M, the orange line) to 2nd Avenue-Houston Street or Delancey-Essex Street.Allen Street runs north to south from Houston Street to just below Canal Street. The 2nd Avenue-Houston Street Station will drop you off a couple of blocks west of Allen Street. The Delancey-Essex Street Station is three blocks east of Allen Street, and three blocks below Houston Street.
Take the Downtown A-C-E (the blue line) from 34th Street-Penn Station to West 4th Street. Transfer at West 4th Street to the Downtown F train to Delancey Street-Essex Street. Delancey Street is one block south of Rivington Street.
Take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to 53rd Street-5th Avenue. Fifth Avenue is one block west of Madison Avenue.
Take the blue subway line (either the C, the A, or the E train) Downtown from 34th Street (Penn Station) to the West 4th Street station. Depending on which exit you use, you will get off at either West 8th, West 4th, or West 3rd Streets, on the west side of 6th Avenue. Cross to the east side of 6th Avenue, and walk south to Bleeker Street. Bleeker Street is one block south of West 3rd. Walk one block east on Bleeker, and you will be on McDougal Street. 102 McDougal Street is a restaurant called Slane's, and it is near the corner of Bleeker. I have included a link to a Google map that pinpoints the exact address you are looking for. The address is marked by a little red bubble with the letter A in it.