In this world the northern kingdoms would be a bit poorer and they couldn't afford as good equipment as empire has, but they have high morale and the population is rather harsh and strong. So what kind of a tactics and weaponry would they use to fight well equipped imperial conquers?

The basic essentials your northern armies would be equipped with are shield (mostly wood, cheap), spear (mostly wood, tiny bit of metal, cheap and relatively easy to use), helmet (metal). The wealthier men would have progressively better body armour but everyone has a shield - the first and most important bit. The semi-wealthy might have leather or padded, the wealthier men might have mail. There's also going to be a lot of older and outdated armour that's been handed down. Now if these people are "harsh and strong" are they regularly raiding and fighting (each other or other states)? Then they might have considerable experience and better gear (acquired from raids). Consider the vikings for equipment wise: large masses of spear armed infantry in a shieldwall. You'll also have the leaders and their retinue who will be equipped with the best gear the leader can afford - fully armoured (in mail suits or coat-of-plates, I'm not sure if plate armour would be available yet in a "poorer" nation).

While it is possible that your average peasant is well trained and other commentators have posted about the English requiring longbow training - note that it was unpopular and as soon as they could ditch it, they did. Peasants had better things to do on their Sundays after working all week. For the most part peasant levies aren't terribly good on the battlefield - look at Hastings where the English shieldwall was broken up when the less disciplined levies chased the retreating Normans downhill and then got butchered. The levies are there to hold, the hard fighting will be done by the professionals - lords and their retainers.

One thing I'll add are town levies. These are likely much better equipped due to the wealth of the town but will likely be restricted in service (based on the town charter). If you're fighting around a major city then you'll benefit from these units that will have a big stake in the outcome. More than some peasant who's been drafted into a larger group of peasants (that he doesn't know) fighting in a county he's never been.

How were the battles fought and what decided who won? Did they just clash each others until the the others felt like they can't win anymore?

Well generally it was easy to decide who won - who ran from the field and suffered the heavier casualties, who still held the battlefield? In the medieval period battles were more rare and sieges more common. A battle is a huge gamble where you risk your expensive army (well, your retainers and their expensive gear) and Murphy's Law could take an easy victory and turn it into a catastrophe. Safer to siege a castle and expand the territory you control militarily. If your army is weaker then you either give up land and play for time (the campaigning season to end, disease to whittle down your opponent, allies to come to your aid, miracles, etc...) If your army is stronger then you obviously want to push until the risks either get too great or the campaign is no longer cost efficient. At the end of the day your retainers are very expensive and a rare resource, losing them will cost a lot of money and time to replace. Losing your peasant levies is less bad but you're still losing workers and will have a harder time getting another levy for the next campaign.

How much morale would affect the outcome of a battle? In this world the northerners are mostly the defending side and they will do anything to protect themselves from the "big bad empire" and the imperial soldiers are far away from home fighting a war that they don't really care about. Does this give a significant advantage to north?

Morale is everything. Armies didn't slaughter each other to a man, no sane person would assent to that. Armies would fight until one side lost the will to continue and ran away - usually it started with one unit fleeing which triggered a panic and routed the remaining soldiers. Better armies will resist panicking while peasant or green levies are more likely to panic and run. The combat model I subscribe to has long bouts of inactivity as the men on each side catch their breath and work themselves up to another round of combat. When one side is ready they push into combat and fight for a couple of minutes until they get tired and then pull back. Most of the participants are fighting defensively, swinging/stabbing their weapons but concentrating on not getting hit. The trained or truly motivated ones (lords and their retainers) will be aggressively fighting to kill their opponent. Actual casualties in this phase will be low, say 5%(ish) percent. This cycle could go on for hours if both sides still have the will to fight. During the lulls the wounded make their way out of the lines. The lines could be pushed back a considerable distance without breaking (obviously being pushed back does cause some fear within the ranks) - Caesar recounts that some Gauls were pushed back 1mi over a battle before they finally broke.

Eventually one side will break. It starts at the rear ranks (which is why you put your 2nd best soldiers in the rear with the best at the front): maybe it's too many casualties suffered, maybe the unit leader got killed, maybe the units been pushed back too far or too often. Regardless the fear sets in that they're going to lose and people start thinking about escape. The guys at the front can't run, they've got people behind them. But the guys at the back will start trying to run away, they don't want to be the last guy standing on the battlefield when the rout happens. The file closer will be trying to hold the rear ranks in place but eventually he catches the panic or they just push past him. This causes a landslide of panic, the guys in front suddenly realize there's no one behind them supporting them and they also panic and run. Men began dropping spears and shields to lighten their load and run faster. Those who kept their wits will retain their gear and try to band together and retreat off the field. The enemy will try to race after them and cut down as many as possible (or capture them if they are well equipped for ransom but no one however is going to ransom a peasant). A good unit will halt pursuit and keep formation but a poorly trained (or poorly led) unit will disperse chasing after the routers - and be very vulnerable to a counterattack. In this situation, the pursuers will often ignore or bypass those who aren't panicking - Socrates survived an Athenian defeat by keeping his head and retreating with arms, the winners ignored him since he was willing to defend himself and concentrated on those who'd thrown away their weapons/shields.

Eventually the winners will halt pursuit but there'll be a trail of bodies going on for a mile or more (longer if the winners have a lot of cavalry). The winners will then recover their wounded, bury/cremate their dead, loot the bodies. The defeated will regroup somewhere but significantly smaller as many of the levy survivors will disperse to their homes.

If a two armies of 10 thousand men would meet in battle how large would the casualties be on average? If I have understood right most of the casualties come when the others decide to retreat, so the losing side would face much larger casualties, right?

Yes. Now casualties will vary depending on a multitude of circumstances: terrain (does it impede the routers or pursuers), cavalry and light infantry (if the defeated side has more of them and they're intact they can shield the routing army, if the winners have more then the defeated are going to have a bad day), time of day (if the rout happened near dusk then the winners won't have much daylight to pursue), and a variety of other factors (e.g. winners too tired to pursue). Pulling some numbers out of my ass (and based on historical battles) I'd guesstimate 500 dead for the victors and 2,000 dead for the losers. Now that doesn't factor in wounded (many of which will die due to infection), captured (more applicable to the wealthier), and those dispersing to their homes. So even if the losing side only suffered 2,000 dead they might only have 3,000 the next day as the rest are too wounded to fight, captured, or the bulk ran off.

I wish you luck on this project. Trying to write fiction with a historical battle is hard, Hollywood has conditioned people to expect confused melees by generic mooks with no self-preservation (aside from the main characters). I'd also add that disease will likely kill off more people than battle, albeit over a longer timeframe. It will target the poor to the wealthy - King Henry V died of dysentary while on campaign. Also please resist the often done trope of bow=gun, they are not similar. Bows (even longbows) are not wonder weapons capable of sniping from 1km, if they were England would have conquered France in the 100yrs War. At Agincourt, the French 2nd wave marched through the arrowstorm and met the English men-at-arms in melee combat (albeit disrupted and attritted by bowmen), they still require men with sharp pointy sticks to win in the end.