Against the Spread tells you how a team plays against the spread - so if you gave them their spread points for each game, what would their record then be? For example (made up numbers):
Week 1: CHI -3 vs BUF. Actual result: BUF by 5. ATS loss.
Week 2: CHI +5 vs SF. Actual result: CHI by 4. ATS win.
Week 3: CHI +4 vs NE. Actual result: NE by 2. ATS win.
So here CHI is 1-2, but is 2-1 against the spread. The SP itself is the spread for that game, the ATS is their overall record for past games when adding in the spread. ATS doesn't really say anything about the current game, unless you believe a team's ATS record is indicative of future performance (hint: it's not in most cases*).
So you'd see something like:
Week 4: CHI @ ATL, SP: CHI -3, ATS: CHI 2-1, ATL 0-3.
That means Atlanta is a Home Dog, Chicago has a favorable record against the spread while Atlanta consistently underperforms (they might win games, but by less than they're expected to, and they lose by more than expected), and Chicago is favored to win this game by 3 points.
Home Favorite means a team favored to win (spread < 0) and playing at home. So yes, a team that is a home favorite, their opponent would be an away dog. This is important to some bettors because some teams overperform at home.
In football betting, being at home is commonly worth around a field goal - so a home team favored by 3 points would be essentially a 0 spread (
pick'em) at a neutral site.
*In some cases, teams are consistently given slightly unfavorable odds and spreads, because they're known to have a lot of betting in their favor - the Yankees, Cubs, Patriots, etc. They might underperform against the spread as a result slightly. However, I wouldn't assume this as a general rule. There's a ton of literature available on things like this if you're interested, but it's way beyond a SE answer.