You are wearing Crown? Or it might be just that some spells cannot be cast in combat?

You've probably run out of mana.

No, it isn't magic points. I can check my stats after I try casting and I have plenty. Spells will fail right after combat starts, with full magic points.

Which spells are you trying to cast in combat?

Tremor, great heal, negate magic, quickness. I can cast great heal the moment i am out of combat, but in combat = failed. Basically, I'm screwed in any tough fight because I can't heal or do anything offensive spell-wise. Scrolls and potions work fine, but not spells.

Sometimes, I try casting great heal and it won;t even ask me who to cast it on. It will say "failed" immediately

Great Heal is noncombat spell, so it will fail in combat.
Also, in combat, sometimes spells will fail (depending of your level vs the spell circle.)

Are spells automatically successful if you are level 5 and are casting a 5th circle spell? Or do you have to be higher level in order for it to succeed every time?

Things that make spells fail:
Character is too low a level to cast the spell (Tremor is seventh-circle, so a sixth-level character would have problems)
Character is casting a spell in a non-compatable situation (Great Heal is noncombat only; Up and Down only work in dungeons; etc...)
Insufficient magic points (spells cost one point per level of the spell; magic points only regenerate in U5 when you sleep)
Spell is not prepared (you haven't mixed it beforehand)
Spell has an illegal target (can't cast magic missiles through walls; can't Heal a dead character, etc...)
Magic is negated (either due to a spell, the presence of certain enemies - do they still do that in U5? - and/or the wearing of Lord British's Crown)
Incantation is wrong (either a typo or simply a misunderstanding)
Character is a fighter-type (no magic points)

I think that's everything. Encumbrance and armour-wearing are not issues for spellcasting in the Ultima series, and Ultima 5 doesn't require a free hand.
So pick your poison. Check each item on the list and see which of the problems you've got.

Thanks for the info. I assumed that Great Heal was anytime. I'll have to check and see the other spells. It might be that they were 1 circle higher. I'm playing mostly from memory from 25 years ago.

You can cast spells while wearing Lord British's Crown. It's just enemies that can't. In other words its effectively a one way and permanent negate magic spell.

Actually world affecting spells like the time stopping spell require you to unepuip the crown. It has something to do with the limited data or something. Then when the spell wears off, you have to reequip the crown to avoid possession, summoning monsters, all the bad stuff high level monsters do.

Well yeah. That's because only one party status effect can be on at any one time. You can find out which is currently active by looking between your food counter and your gold. I'm not sure if wearing the crown will cause any party buffs to fail, though. They might do, but they also might just replace the crown's effect. You can cast all other spells though while keeping the crown's effect.

Here's another question: do rooms in dungeons ever respawn, or do they stay cleared after you kill all the monsters in them? If I leave a dungeon and go back in, the rooms are still empty.

That's a new feature in Ultima V. When you clear a room it stays cleared forever. If you only partially clear a room it will be repopulated again when you re-enter it. You need to fully clear it out to keep it cleared. Some rooms have secret enemies hidden behind walls to stop you from being able to clear the room out entirely. If you use a morningstar or halberd you can attack them over the walls to clear these rooms out too.