For the specific example you gave:
Phaeston IV deals 37.2 damage, 372 DPS w/o reload, 292 DPS w/ reload
Avenger IX deals 47.1 damage, 393 DPS w/o reload, 321 DPS w/ reload
This is without attachments or damage improving abilities taken into account, however, since they improve damage by a percentage, the ratio between their DPS will remain the same.
If their levels were equal the Phaeston would have a slight DPS advantage.
If you want to make other specific comparisons you'd have to use the data in the spreadsheet and calculate the exact stats, since a spreadsheet for all weapon levels would be huge and impractical.
Here are the general improvements from weapon upgrades:
Level 10 weapons deal 25% more damage than level 1 weapons, so weapon damage improves by 2.(7)% per level. This applies to all weapons in single player, and with the exception of a few errors, all weapons in multi player as well.
Weapons with more than 40 spare ammo at level 1 improve it by 2.(7)% per level(25% total), and the ones with less improve it by 1 per level and 2 at level 10(10 total).
The Valiant is the only weapon that does not improve ammo reserves(likely an error and an intended 20 to 30 spare ammo capacity).
Weight upgrades are not consistent among all weapons and have several different "categories":
Level 1 weight - level 10 weight - upgrade per level - weapons in category
250% - 200% - 5.(5)% - Black Widow, Widow, Javelin, Claymore, Crusader
200% - 140% - 6.(6)% - Falcon, Revenant, Saber, Argus, Graal, Geth Plasma Shotgun, Raider, Valkyrie, Collector Assault Rifle
175% - 100% - 8.(3)% - Mantis, Indra, Chakram, Valiant, Raptor
150% - 90% - 6.(6)% - Geth Assault Rifle, Scimitar, Incisor, Talon, Scorpion, Arc Pistol, Katana, Eviscerator, Wraith, Mattock, Phaestom, Viper
100% - 50% - 5.(5)% - Disciple, Hurricane, Vindicator, Avenger, Hornet
75% - 35% - 4.(4)% - Paladin, Carnifex, Tempest, Locust
60% - 25% - 3.(8)% - Phalanx, Eagle
50% - 20% - 3.(3)% - Predator, Shuriken
These are the single player stats. For up to date changes of weapon stats in multi player visit http://social.bioware.com/forum/1/topic/343/index/9544143/1
So it's between 3.(3)% and 8.(3)% per level
Shepard has 75% starting weight capacity, which can be increased by upgrading his passive skill(adept and engineer up to 125%, soldier up to 175% and the rest up to 145%).
Formula for recharge speed from weight is:
200% + weight capacity - weight of all carried weapons
It's capped at +200% and -200%(other recharge speed bonuses do not adhere to this cap).