In Dark Souls 2 you can consider infusions and the +N upgrades as two separate properties of the weapon. For example, if you have a +10 Claymore and infuse it with lightning, you'll have a +10 Lightning Claymore. Further, you can "uninfuse" it and it'll go back to just a +10 Claymore. There's no upgrading of the infusions themselves. That said, if you infuse a less than +10 weapon, you can still continue upgrading it to +10 with titanite, and it will retain its infusion as you upgrade it.
Whether or not to infuse a weapon is another question. Each infusion does something different to the weapon. Something like Lightning will take away some of the base damage on the weapon and replace it with lightning damage. This also affects the scaling. A purely strength based character probably wouldn't want to infuse with lightning, since doing so would likely lower the scaling off of strength and add scaling for lightning (which is affected by Faith). A Strength/Faith hybrid build might do a lightning infusion, though.
There's so many combinations, and each type of infusion works differently (Raw increases base damage but lowers scaling, Mundane adds scaling based on your lowest stat, etc). My best advice would be to experiment with different types, as there's not necessarily a "right" answer (keep in mind that you can preview the changes before you actually go ahead with the infusion). It depends a ton on your build and playstyle.
Here is a reference from the wiki
Starting Class Sorcerer
Level 150
Vigor 20
Endurance 15
Vitality 15
Attunement 40
Strength 3
Dexterity 7
Adaptability 10
Intelligence 46
Faith 47
Weapons
Right Hand:
Dark Sunset Staff +5
Binoculars
Left hand:
Dark Chime of Want +5 (short range)
Lightning Dragon Chime +5 (long range)
Equipment
Hexer's Hood +5
Hever's Gloves +5
Velstadt's Armor +5
Velstadt's Legs +5
Third Dragon Ring
Chloranthy Ring +2
Clear Bluestone Ring +2
Southern Ritual Band +2
Spells
Great Lightning Spear x2 (PvE) -or- Sun Spear (PvP)
Dark Orb x2 (PvE) -or- Resonant Soul (PvP)
Scraps of Life
Dark Fog
Dark Storm
Soul Greatsword (If you have the slots, depends on whether you got the +2 southern ritual band)
Tips
PvE tactics for this build are fairly straightforward. Utilize your bottomless pit of dark orbs to ruin everything in front of you. With these stats and this equipment, each orb will do in the ballpark of 500-800 damage depending on the enemy, and you'll be capable of casting one every 1.5 seconds. You have plenty of other tactics to use, like using scraps of life to maintain separation on a horde of enemies, or using your other bottomless pit of lightning spears to snipe enemies from a safe distance. Dark Fog is also very very useful in certain situations, mainly against stationary targets like enemy mages or shield using slow opponents. For example, poisoning a Syan Knight will kill it outright, just wait a minute. Dark storm is fairly situational in PvE, and is best used in coop. This spell is also the reason you want to be wearing heavy armor, because you are nearly guaranteed to take hits while using it, but the damage an enemy will take while attacking you is almost always worth the trade (it generally kills things outright).
PvP requires an extension of the PvE tactics above, but you must be much more savvy in your casting, and faster. Success is rarely instantaneous (no crazy one shot backstabs in this class), but you can almost always impose your will and win a protracted battle. Using spells in combination is key, and you must always keep the pressure on. If you're facing a heavy melee opponent, using scraps of life to maintain separation is extremely important. If you manage to score a hit (which will knock them down) follow up immediately with dark fogs to inflict poison. Or, if you're facing a fast melee opponent, bait them in with dark storm, and follow immediately with a soul greatsword if they are smart enough to not rush into the storm. If the fight is long range against a mage or archer, Dark fog followed by sun spears usually wins the fight quickly. You may want to substitute the Sun spears for Crystal Soul Spear, as that seems to land hit more often in PvP.
Best Answer
No, the Avelyn is found in a metal chest and Bonfire Ascetics don't reset those. The only way to get a second Avelyn from this chest is in NG+ or you could have another player drop one for you online.
Another way to get more Avelyns is in the Doors of Pharros area (Brightstone Cove Tserolda in SotFS), by killing the red phantom Bowman Guthry, who only appears if you play offline and can repeatedly be killed by using Bonfire Ascetics.