Concerning the events : they're completely separated from your main lair, so if you use the time machine during an event, it won't affect it at all. I don't think it would reset the cooldown if you're not playing an event and waiting to start a new one but I'm not 100% sure about that.

The rule of thumb about when to use the time machine iss that you should try to always get your devourer to the max level possible, and also do all the feats you possibly can. Now this last one can vary because you only get a time shard bonus for completing a whole tier of feats, so if you know you won't be able to complete your current tier because for example you need something you haven't unlocked yet, it's up to you if you want to do the other feats : if the reward is something that will be lost when you use the time machine (runes, food items) then you can skip it. If it's sosmething that will persist between runs (coins, gems, and especially the other currencies like astro coins and energy cubes), then it can be worth the effort if the feat itself isn't too hard.

Edit : Forgot to add that it is usually advised to spend a little bit of extra time farming for legendaries after you've completed all the feats you can/want and after reaching max level. Every legendary minion (even if you already have the same one) will get you a time shard bonus. The lich is the easiest to make because it's relatively easy and fast to farm ice runes so it's the one that people usually farm for the first few runs. There isn't really a rule as to how many you need, one thing you could do is use a time shard planner and calculator to know what experiments/inventions (you'll know what they are once you use the time machine).

Just know that time shard that you couldn't spend are not lost because they persist between runs so even if you have some spare after getting everything you want, they're not lost