Good thing you didn't raise your handicap, the only way it goes is a higher penalty against you.
Combat occurs on the square your unit ends at, where you move from doesn't impact it at all. Also, where your unit was before it fights doesn't matter - units basically teleport into their new square, if enemy units are going the other way you'll just switch places with no combat occurring. The only time this isn't the case is when a nuke is used, and that destroys units before they get a chance to move out.
If your drop pods are being blown up, it's because the AI is detecting them. Try to drop where he won't have units nearby, and always drop into mountains. Then scatter your forces at least into 2 trooper 1 tank combinations so you won't get blown up the turn after.
If you haven't looked at it, the expert's guide on the wiki can be very useful for figuring out how to win battles - note the combinations that do well against the ones the AI usually uses, and try to hit with those. Particularily useful against the AI are 2 troopers 1 tank, and 1 trooper 1 bomber. The expert's guide can be found at
http://wiki.conquest-game.com/Expert%27s_GuideIf you want some more basic ideas, you could also read the beginner's guide on the wiki, at
http://wiki.conquest-game.com/Beginner%27s_Guide For my best guess as to why you're having problems, you're probably not being aggressive enough with your units. Don't leave any except the basic trooper (Or tank, as the situation is) guarding a city - move them out towards the enemy to keep fighting. Try to prevent him from even getting close enough in the first place to attack your cities, rather than trying to hold off at your cities.
Hope this helps!