Step 1: The second clue in A1 suggests that the Kansas City Royals are E4. Kings & Queens are considered "Royalty".
Step 2: The clue in E4 implies that the Philadelphia Phillies are C5. The (Phillie) Phanatic is indeed the Phillies mascot.
Step 3: The clue in C5 explains that the Minnesota Twins are A2. C5 also explains that column 5 is in reverse alphabetic order. That clue along with the first clue in A1 (New York teams are in the corners) implies that the New York Yankees are in A5 (Mets cannot be above the Phillies). This also implies that the New York Mets are F1.
Step 4: The second clue in A5 explains that the Arizona Diamondbacks are B2. Diamondback rattle snakes are reptiles, and happen to be the only reptile mascot in the MLB.
Step 5: In order to move further along from here one must know all the California teams.
Los Angeles Angels
San Francisco Giants
Oakland Athletics
San Diego Padres
Los Angeles Dodgers
Since there are 5 California teams and since the first clue in A5 (California teams all share a row) they must all reside in Row D. No other row has 5 available spots as shown above. Now, the first clue of F1 says the Angels are in column 1. That clue in addition to finding out all California teams are in row D makes the Los Angeles Angels in D1.
Step 6: Clue D1 explains that all stadiums named after brewing companies are in row F. The teams with stadiums named after brewing companies are as follows:
Colorado Rockies (Coors Field)
St. Louis Cardinals (Busch Stadium)
Milwaukee Brewers (Miller Park)
Now referencing clue B2 (All US birds are in column 5). The bird teams are:
Toronto Blue Jays
Baltimore Orioles
St. Louis Cardinals
The Toronto Blue Jays are cross out in this list above because they are not in the US, however the other two bird teams are. So, from clue D1 and clue B2 the St. Louis Cardinals must be F5.
Step 7: The second clue from F5 along with clue D1 implies that the Colorado Rockies are F4. Now, the Orioles must be in column 5 from clue B2, and since the first clue from C5 says that column 5 is in reverse alphabetical order, the Orioles cannot be in B5 (O is before P). They also cannot be in D5 because they are not a California team, therefore the Baltimore Orioles are E5. Now, the only California team who is alphabetically between the Orioles and the Phillies are the Padres, so the San Diego Padres are D5.
Step 8: Clue D5 says that teams with the same city name touching each other in this row, the only California teams that share a city name are the Angels and the Dodgers, so the Los Angeles Dodgers reside in D2. Clue E5 explains that the Reds are in row A. Clue F4 indicates that one of the Twins divisional opponents are bordering them in row A. The Twins division (AL Central) includes the following teams (Teams already used are crossed out):
Kansas City Royals
Chicago White Sox
Cleveland Indians
Minnesota Twins
Detroit Tigers
The Atlanta Braves in A1 are not a division rival of the Twins, so their division rival must be in A3 for clue F4 to hold true. Since the Reds are in row A, but they are not a division rival of the Twins (cannot be A3), the Cincinnati Reds must be A4.
Step 9: Clue A4 indicates that the Boston Red Sox are B4 ("Reds"ox). Clue D2 explains that none of the adjacent cells to the Dodgers (C1, C2, C3, D3, E3, E2, & E1) are one of their divisional opponents. The Dodger's division (NL West) includes (teams already used are crossed out):
Los Angeles Dodgers
Colorado Rockies
Arizona Diamondbacks
San Diego Padres
San Francisco Giants
The only team that is left without a home in the Dodgers's division are the Giants. Since they are a California team they must be in row D as stated previously and with clue D2 indicating the Giants cannot be in D3, this implies that the San Francisco Giants are D4. This indicates from looking at the list of California teams in Step 5, that the Oakland Athletics (last California team) are D3.
Step 10: Clue A2 (socks come in pairs) suggest that the White Sox are either in B3 or B5. The second clue of F1 (B3 and C3 start with "M's") eliminates the White Sox from B3, therefore the Chicago White Sox are B5. This also holds the first clue of C5 true since W is in between P and Y alphabetically.
Step 11: Clue F4 (Twins division rival borders them) is down to only two possible teams (Indians and Tigers). All other divisional teams have already been used (AL Central team list is in step 8). However, clue D4 indicates Justin Verlander's teams are in column 1. His teams are:
Detroit Tigers
Houston Astros
New York Mets
Since the Mets are already in column 1 and the Tigers have to be in column one they cannot be in A3, therefore the Cleveland Indians are A3. The second clue from F1 says that B3 and C3 start with the letter M. The letter M teams in the MLB are (teams already used are crossed out):
Seattle Mariners
New York Mets
Miami Marlins
Clue B5 says that the Florida teams are in row B. The Florida teams are:
Miami Marlins
Tampa Bay Rays
Since there are only two spots left in row B and B3 must start with the letter M the Miami Marlins are B3 and the Tampa Bay Rays are B1. This also indicates that the Seattle Mariners are C3.
Step 12: Clue A3 (No Texas team is in row E) along with D4 (Verlander's teams are in column 1) implies that the Houston Astros are C1, since they cannot be in the only other available spot in column 1. This means the Detroit Tigers are E1. Clue B4 indicates that A3 and A4 share something geographically in common with C4 and C5. Since the Reds & Indians are from the same state (Ohio) it implies that the Pittsburgh Pirates are C4. The Pirates and Phillies share the same state (Pennsylvania).
Step 13: Clue C4 suggest that the Nationals (Washington D.C. is the capital of the US) and the Blue Jays (from Canada) are next to each other. They cannot be in row F because one of those spots are reserved for the Brewers (clue D1). This means they must be in row E since it is the only row with two spots available. Clue D3 (colors cannot be in this row) implies that the Toronto Blue Jays are E2 and therefore the Washington Nationals are E3. Clues C1 indicates that the Astros in-state rival is in column 2, but not touching them. This means the Texas Rangers are F2.
Step 14: Clue D1 (stadiums named after brewing companies) implies that of the two remaining spots, the Milwaukee Brewers are F3. Lastly, this leaves the Chicago Cubs in C2. Which holds the first clue of F5 true.
HOORAY! The puzzle is complete.