Should I build damage when I am the only tank/Front line?

As a Elise the only thing you really want to build for damage is magic-pen, your base damage and missing-maximum target hp scaling on Q just too good by themselves and you got way more from penetration then from AP. That makes Elise such an amazing champion for being ahead as while building defensive you still get really good offense.

Other advice (especially for solo-q) as a jungler you never want to try "save" lane. You may gank such Poppies as much as you physically can, but if their opponent is so much better - any advantage you gave would be lost next moment you leave a lane, and in the end of the day you just waste your time to help helpless person. As was mention before - your goal is objective control and giving initial advantage. If you see that one lane is struggling proper reaction would be not try to babysit that lane, but apply more pressure to others trying to force such Jaxes to start roam asap and stop snowballing in their warm and cozy lane which they already dominating, and will give opportunity to free farm to your teammate to catch-up.

In cases if you're only tank in team, you should not to try to fight their front-line and/or peal for backline - it would be a support's task in such cases (most not-tanky supports as Nami could peal and keep adc alive way better then you can). If you're the only tank - you go for their backline and try to win as much time as you can, and depending what king of jungler you play - ether assassinate their heavy-hitters or get as much cc on them as you can and stay alive.

Its usually always good to go one damage item. For Elise it isn't always a necessity to build a damage item but it sure does help. Listen to the pros when they talk about when to build damage when you're behind so you can possibly get a pick. If you're ahead you want to build tankier and more sustain so you easily out last your opponents.

Did I make a mistake building pure tank? Probably, you need to build some damage so that you can be seen as some threat so that you aren't ignored past your CC.

Should I build damage in a circumstance like this in the future? I would have gotten at least a Lliandry's. Maybe a void staff also if they had high MR.

How do you make the call to gamble and build offensive when you up against a fed team without a proper front line to soak up and block off their damage and dives? Normally you don't. You can hope for some picks to pick off some targets and hope that the rest of team doesn't collapse on you. Most of the time though you still need a tank to soak up damage/attention.

When you are the only frontline, you never build more than 1-2 damage items. So Elise in particular the only damage item you should need is a Lliandy's. Void staff if people are stacking too many resistances against you but everything else should be tanky. It's your job to either initiate and either kill the enemy carries or keep them busy a long time.

Some frontlines synergize and even do increased damage with damage items. Malphite does more damage the more armor you have. Champions like this, you will build even tankier than others since building tankiness gives you a slight increase to damage also. Mundo is sort of an exception since he doesn't really scale damagewise off tank items (besides thornmail). You build him pure tank anyways because his base damages are high enough to keep the enemies attention normally.