No, you don't pay any extra upkeep based on if your horse dies.
With most builds, I get 15 str before getting any agi. With cav I get 12 str and then dump points into agi, usually until I'm 12str/15agi. Then I put points into str again until I'm at whatever str I wanted to end up with, and then go back to agi. I've found this makes for the most fun leveling. You get 4 or 5 ps quickly and feel useful and get kills. Then you get the agi and actually keep up with people and are able to run away.
What weapon you start with all depends on what you're going to use on horseback. If you want to be a pure lancer with all wpf in polearms, then obv get a polearm. Warspear is great. If you're going to use a sword while mounted too, then use a sword when you're on foot.
You're using a build calculator, right? They are very useful when making a character.