Valour being equal, Swords beat spears
Valour and class and morale and fatigue as KOTR says being equal, Yes - dont matchup feudal men at arms with chivalric seargents and excpect a win. Swords get a hidden +1 attack against spears in VI - this was supposingly to restore the RPS as v1.0 MTW was ver much a spears game (swords were essentially flankers at best).

Spears beat cavalry.
Yes and no. If you match up vanilla spearmen against knights the spears certainly lose. Feudal and chivalric seargents usually persevere to victory up unless their morale breaks - most spears suffer from low morale in MTW (= 0). Few spears have good morale such as, methinks Italian Light Infantry (= 2).

In MTW spears have very low attack and good defense and in addition get similar bonuses versus cavalry that is again good defense and low attack with rank bonuses. This, and their immunity to charges makes them good for holding cavalry. However they often dont have the power to deliver kills, it takes ages for them to kill the cavalry while on hold formation. if you are confident that they wont break (say because your battle line is in a crisis) and they are facing reasonable opponents for their class (not vanilla spears against chivalric knights) then put them on engage at will - their offensive ability is best. However its best receiving charges and pinning superior opponents with hold formation - their low morale otherwise betrays them.

The only TW game that spears worked properly is STW - they were also 60 men units but were eating cavalry alive - the only working total war RPS to date.

Cavalry beat swords.
Yes and no. Heavy cavalry does. Medium cavalry might with the charge. Light cavalry never frontally always flanks while engaged. However MTW has a fine selection of heavy horse pounders for all factions. In terms of match up these should be dealing with enemy swords while ideally your spears keep enemy cavalry occupied and your swords beat their spears. If you manage this its battle over very quickly.

Ranged units are auxiliary with few exceptions.
Yes, as the cards often say keep them out of melee. In practise use them only as a last reserve frontally or for flanking when its safe for them.

Heavy cavalry beats light cavalry.
Frontally yes, with no exception that i know off - perhaps camels in the desert might beat some heavy horse sometimes.

Beserkers are fun.
Fun is subjective but yes.

!it burnsus!