Five wickets in a Test innings against country of birth

7-55 TK Kendall for Australia v England at Melbourne, 1877
5-78 WE Midwinter for Australia v England at Melbourne, 1877
6-120 WH Cooper for Australia v England at Melbourne, 1881
5-64 J Middleton for South Africa v England at Port Elizabeth, 1896
5-51 J Middleton for South Africa v England at Johannesburg, 1899
7-63 AE Hall for South Africa v England at Cape Town, 1923
6-82 AE Hall for South Africa v England at Johannesburg, 1923
6-100 AE Hall for South Africa v England at Johannesburg, 1928
5-36 GOB Allen for England v Australia at Brisbane, 1936
6-49 GS Ramchand for India v Pakistan at Karachi, 1955
6-77 DE Malcolm for England v West Indies at Port-of-Spain, 1990
6-111 CC Lewis for England v West Indies at Birmingham, 1991
5-32 AR Caddick for England v New Zealand at Birmingham, 1999
6-122 AR Caddick for England v New Zealand at Christchurch, 2002
6-63 AR Caddick for England v New Zealand at Wellington, 2002
5-32 Imran Tahir for South Africa v Pakistan at Dubai, 2013
Players born before partition in what is now Pakistan are regarded as being born in Pakistan

Test debut on birthday

BB Cooper (33) for Australia v England at Melbourne, 1877
FGJ Ford (28) for England v Australia at Sydney, 1894
FW Tate (35) for England v Australia at Manchester, 1902
AER Gilligan (28) for England v South Africa at Johannesburg, 1922
BJ Sealey (34) for West Indies v England at The Oval, 1933
J Iddon (33) for England v West Indies at Bridgetown, 1935
DG Phadkar (22) for India v Australia at Sydney, 1947
SB Joshi (27) for India v England at Birmingham, 1996
RD Jacobs (31) for West Indies v South Africa at Johannesburg, 1998
RL Powell (21) for West Indies v New Zealand at Hamilton, 1999
S Chattergoon (27) for West Indies v Sri Lanka at Port-of-Spain, 2008
Shan Masood (24) for Pakistan v South Africa at Abu Dhabi, 2013
Cooper's birthday was on Test cricket's birthday

Century and hat-trick in same first-class match

G Giffen 13 & 113, 6-55: Australians v Lancashire at Manchester, 1884
WE Roller 204, 4-28 & 2-16: Surrey v Sussex at The Oval, 1885
WB Burns 102*, 3-56 & 2-21: Worcestershire v Gloucestershire at Worcester, 1913
VWC Jupp 102 & 33*, 6-61 & 6-78: Sussex v Essex at Colchester, 1921
RES Wyatt 124, dnb & 5-39: Marylebone Cricket Club v Ceylonese at Colombo, 1927
LN Constantine 107, 7-45 & 6-67: West Indians v Northamptonshire at Northampton, 1928
DE Davies 139, 4-27 & 3-31: Glamorgan v Leicestershire at Leicester, 1937
VM Merchant 1 & 142, 3-31 & 0-17: Dr CR Pereira's XI v Sir Homi Mehta's XI at Bombay, 1946
MJ Procter 51 & 102, 3-43 & 5-30: Gloucestershire v Essex at Westcliff-on-Sea, 1972
MJ Procter 122, 0-32 & 7-26: Gloucestershire v Leicestershire at Bristol, 1979
KD James 103, 5-74: Hampshire v Indians at Southampton, 1996
JEC Franklin 14 & 109, 3-59 & 1-28: Gloucestershire v Derbyshire at Cheltenham, 2009
Sohag Gazi 119 & 21, 2-126 & 7-79: Barisal Division v Khulna Division at Khulna, 2012
Sohag Gazi 101*, 2-79 & 6-77: Bangladesh v New Zealand at Chittagong, 2013

Roller is only one with 200
James is the only one with 4 wickets in 4 balls
Gazi joins Procter as the only ones to have done it twice