If I upgrade a weapon to +10, can I change it to a quality weapon and upgrade it to +5?

This is about Demon's Souls on PS3. If I upgrade a weapon to +10, can I change it to a quality weapon and upgrade it to plus 5? Also, do I keep the base stats from the +10 and does it add to the +5.

For example, Longsword +10 to Quality Longsword +5. Do I keep the stats for the base +10 or do I have to choose to upgrade to a quality weapon before plus ten? What I mean by that is can I change it at +10 or do I have to do it before the base weapon maxes out at +10?

No, you cannot upgrade from regular +10 to Quality.

You can only switch to Quality upgrade path (using Clearstones) from +3 Normally upgraded weapon (get there using Sharp-/Hardstones).

The only way to get to Quality from +10 Normal is to use Meltstone to actually rollback weapon to basic +0 state, which is extremely wasteful since you won't get any original materials back.

As for weapon stats, you will lose some of the raw damage, but it should be compensated by better DEX/STR scaling. If we take Longsword for example, Normal +3 has 104 Base Damage, but Quality +1 has only 92.

See more on this "Upgrades" wiki page, it includes nice upgrade path charts.