You will get most of these while working on getting the presents, cousins and 100% Collection achievements, but once you're done with that if you still have not done 200 requests, just simply play Egg School (Tutorial level) over and over until you get it. You can do any request, but this is the shortest one.

Note: When you check the "Everyone's Katamari" section in space, this total refers to Eternal Modes and Time Attacks as well as your Normal Modes, so you will most likely need to get that number higher than 100/200, but it should give you an idea of how far you are.

There are 3409 items to collect in the game. You can check your progress from the tree on the main island. They are split into different groupings. The best thing to do is unlock Eternal Modes. That way, you can roll the levels in all different ways and not worry about getting the size requirements. Once you're bored or too big to pick anything else up, simply hit  to save and exit the level.

Some things you can no longer get credit for if you are too large, so try to move to new areas as soon as you can, as well as replaying levels so you can roll different areas at different sizes. Some things only show up in certain levels (even if the levels start you in the same area, there will be different people and things lying around), so make sure you play all the levels over again.

Note: Seadome and Schloss Cosmos seem to have most of the items, so play those often.

Once you finish a level, it will open the Time Attack mode for that level. You can't fail these, so simply finish each Time Attack and you'll get the achievement.

Note: There are only 10 Time Attack levels. Egg School, Roller Roaster and Chateau Notre Desir do not have Time Attack modes.