Do non-proc-shots fired by elemental weapons still carry the weapon's element?

Question

TL:DR: If my slag weapon procs its slag effect, should I switch to another non-slag weapon or will my majority of non-proc-bullets be sufficient to make use of the slag.

If a weapon is of the Slag element, does this make every bullet it fires of said element, or just the shots that proc the effect?

In field concept, say I'm using this here rifle:

If I shoot a critter with the above weapon, and I shoot a burst of say 10 rounds, (ignore the mag size you see above) assuming round #4 procs the slag effect and is the only round to proc the effect, will the other rounds fired after round #4 do extra damage due to their not being slag, or will they count as alternate sources of slag despite not proc-ing the weapons effect.

Answer 1

If a weapon is of the Slag element, does this make every bullet it fires of said element, or just the shots that proc the effect?

Judging from the damage-numbers colour, if a weapon is of the Slag element, this makes every bullet that it fires of said element. The same goes for the other elements, except for explosive, which always procs. 
 

If I shoot a critter with the above weapon, and I shoot a burst of say 10 rounds, (ignore the mag size you see above) assuming round #4 procs the slag effect and is the only round to proc the effect, will the other rounds fired after round #4 do extra damage due to their not being slag, or will they count as alternate sources of slag despite not proc-ing the weapons effect?

If you shoot a critter with your illegitimate weapon, and you shoot a burst of say 10 rounds, (ignoring the mag size I see above) assuming round #4 procs the slag effect and is the only round to proc the effect, the other rounds fired after round #4 will not do extra damage due to their not being slag. I'm not sure what you mean by "count as alternate sources of slag".

Exception:

In Ultimate Vault Hunter Mode, [...] [s]lag on slag damage is also increased by 50%.

http://borderlands.wikia.com/wiki/Slag#Effectiveness
"By" is probably supposed to be "to", because 0% * 150% is still 0%.
 

TL:DR: If my slag weapon procs its slag effect, should I switch to another non-slag weapon or will my majority of non-proc-bullets be sufficient to make use of the slag.

If your slag weapon procs its slag effect, you should switch to another non-slag weapon. Your majority of non-proc-bullets will not be sufficient to make use of the slag.
 

Thanks to galacticninja for the extra information. ;)

Answer 2

Elemental weapons deal damage on hit and also have a chance of triggering a status effect per hit (this is a damage-over-time effect for non-slag weapons). An exception are Explosive weapons, which don't have DoT and always trigger their Explosive splash damage - more info here).

A slag weapon like in your example will always deal slag damage on hit, and will also have a chance to trigger the slag status effect on the enemy (or "slag" an enemy) per hit.

In Ultimate Vault Hunter Mode (UVHM), slag on slag will deal a 50% damage bonus. In your example, if "round #4 procs the slag effect and is the only round to proc the effect", rounds 5-10 will gain a 50% damage bonus while the enemy is still under the slag status effect. This makes slag weapons more useful in UVHM. However, this is not the case in non-UVHM playthroughs (slag on slag will deal no bonus damage).

It is preferable to switch to any non-slag weapon once your slag weapon has slagged (procs the slag effect on) an enemy, as the bonus damages for hitting a slagged enemy with non-slag damage are 100% for non-UVHM, and 200% for UVHM.