The two blanks can be filled by:
Ethiopia and Guatemala.
Let's dissect the patterns here...
The first pattern to note is that the last letter before 'ing' in the activity name is the Roman numeral of the number of the attempt: skIing (1), diVing (5), boXing (10), tiLing (50), raCing (100), aiDing (500), gaMing (1000). However, this is purely for construction purposes - it plays no real further role in the solution process...
What is significant, however, is the relationship between (i) the source word for the activity without the 'ing' (adding an 'e' when necessary, for DIVE, TILE, RACE and GAME), (ii) the capital city of the country, and (iii) the time. Specifically, in each instance: when you remove the letters of the source word from the name of the capital city, you are left with the same number of letters as the time mentioned. Like so:
Skiing, Finland, 5 o'clock: HEL[S]IN[KI] - SKI = 5 letters;
Diving, Uruguay, 6 o'clock: MONT[EVID]EO - DIVE = 6 letters;
Boxing, Luxembourg, 7 o'clock: LU[X]EM[BO]URG - BOX = 7 letters;
Tiling, Honduras, 7 o'clock: [TE]GUC[I]GA[L]PA - TILE = 7 letters;
Racing, Australia, 4 o'clock: [C]ANB[E]R[RA] - RACE = 4 letters.
What we therefore need to find are...
...countries whose capital cities leave 7 and 9 letters once AID and GAME are removed from their names, respectively.
The only sovereign states which satisfy these criteria are:
ETHIOPIA: [AD]D[I]S ABABA - AID = 7 letters;
GUATEMALA: [G]U[A]T[EM]ALA CITY - GAME = 9 letters.