1.) What kind of pain does Hell involve? Is it physical or emotional/mental? Did not Pope John Paul II describe it as being completely and eternally separated from God?
Yes, and that complete and eternal separation from God causes pain in every fibre of our being - mental, emotional and physical - since it is separation from the source of physical as well as mental/emotional life and wellbeing.
2.) Can you go to Hell without forgiven mortal sin from the Church?
The question is confusing. If you’re Catholic and in a state of mortal sin, then you must, presuming you are able, go to confession and have those mortal sins forgiven. That is the ordinary way they are absolved.
There is such a thing as Perfect Contrition, whereby one who is genuinely sorry for their sins out of love of God rather than mere fear of hell or any other motive, AND who resolve to confess their sins when they are able to do so, will be forgiven and saved even if they are unable to get to confession prior to death.
Note that perfect contrition on its own does NOT mean one can receive the Eucharist or other sacraments without prior confession - apart from a few very exceptional circumstances. And note that one is at all times obliged afterwards to confess as soon as one is able. Note too that the requirement to sacramentally confess applies only to Catholics - of course, since sacramental confession is available only to them
3.) If Purgatory does not exist in time, then why has the Church issued partial indulgences with periods of time?
To understand this, you need to understand that centuries ago the Church handed out what were called ‘canonical penances’. In other words for your sins you’d probably be required to spend a certain number of days, months or years doing a set of standard penitential activities, such as fasting.
The ‘300 days/500 days/7 years’ indulgence simply means that saying that particular indulgenced prayer, or doing that particular indulgenced action, would have the same benefit as spending 300 days/500 days/7 years doing ‘canonical penance’. The amount of purgatory that would be ‘worked off’, if you will, would be the same amount as that ‘worked off’ by the period of penance.
Needless to say, being that ‘canonical penances’ are largely a thing of the past, the whole concept of a time period attached to indulgences became hugely misunderstood and people literally thought so many days would come off one’s ‘time served’ in purgatory as if it were a jail sentence.
Hence in modern times indulgences are simply either ‘partial’ or ‘plenary’, with no particular number of days/years attached. Avoids the confusion.