English grammar

1. The players try to make as little contact as possible.. .[etc.]

2. It has been difficult to integrate all the groups in our society . . .[etc.]

3. [I see nothing wrong with that sentence]

4. Sy's doctor suggested that he take his medicine ...[etc.]

In #4, Sy's is correct. The general rule is that if a word or name ends with an 's', there is no need to add another 's' to make it a possessive. If the name were Charles instead of Sy, you could write "Charles' doctor suggested..." etc., although Charles's would not be incorrect either. But you never indicate possessive by just adding an apostrophe (') where the name ends in a vowel or any letter other than 's'.

1. The players try to make a(s) (few) contact(s) (as little contact) as possible with the opposing team to avoid penalites.

2. It has been difficult to integrate all the groups (to) our society due to their different social values and traditions. This is a tough one Andy because I think he's trying to express the idea that these particular outside groups are hard to integrate within his group because their values are so different, hence the "to". However I could be spitting hairs

3. Peter told us that he (had)(finished) his project before leaving the house yesterday morning. Don't ask me why

4. Sy's doctor suggested that he (takes) his medicine with water 30 minutes before eating.

He's quite right, it's "take" but I'm not sure Andy has quite covered every base. For instance,

Sy's doctor suggested that he (takes) his medicine with water 30 minutes before eating.

Of course we realize that Sy's doctor supposedly isn't suggesting Sy emulate his (Doc's) example but subtleties like this have started wars. Yea, Jang, you might conceivably read it either way