what does "buy a very mixed bag" mean?
【Even if they all went all out, Yarborough told Mac Kilduff and Congressmen. Joe Kilgore, John Young, Tiger Teague, and Henry Gonzalez, the outlook was bound to be cloudy. He invited them to take a look at the Mexican vote. Mexicans were volatile—Gonzalez nodded in agreement—and with himself and Connally on the same ticket, they would be asked to buy a very mixed bag. Some would shy away. A lot of others would split their tickets.】

1.what does "buy a very mixed bag" mean?
2. does "shy away" mean "avoid voting?"
3. does"split their tickets" mean"to destroy their tickets?"or , does "split their tickets" mean "break up their tickets, half to Connally, and the other half to Yarborough'

Question

what does "buy a very mixed bag" mean?
【Even if they all went all out, Yarborough told Mac Kilduff and Congressmen. Joe Kilgore, John Young, Tiger Teague, and Henry Gonzalez, the outlook was bound to be cloudy. He invited them to take a look at the Mexican vote. Mexicans were volatile—Gonzalez nodded in agreement—and with himself and Connally on the same ticket, they would be asked to buy a very mixed bag. Some would shy away. A lot of others would split their tickets.】

1.what does "buy a very mixed bag" mean?
2. does "shy away" mean "avoid voting?"
3. does"split their tickets" mean"to destroy their tickets?"or , does "split their tickets" mean "break up their tickets, half to Connally, and the other half to Yarborough'

Answer

1. "A mixed bag" means a collection of different things. 
2. "Shy away" in this context, is when the voters would shy away from those politicians  - yes, perhaps they wouldn't vote at all, or perhaps they would vote for someone else. 
3. A "split ticket" is when a ballot (the voting form) is for more than one election (e.g. national and local elections) and the voter chooses candidates from different political parties - this has happened a lot in United States politics.