Short rests require an hour in which
a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
(PHB, page 186.)
Eldritch Master takes one minute, and doesn't specify what you can and can't be doing in that time. It's unclear (i.e. open to DM ruling) if you have to spend that minute doing nothing except entreating your patron, but even if that's true, sometimes, you don't have an hour. If you're running away from something, you might just be able to get a minute lead on it to quickly get your spells back. An hour lead is a lot less likely.
A minute is a very short time out of combat, and even in combat, it's 10 rounds. In a protracted fight you could probably pull this off. In a really protracted fight, you might need to. For example, if you're fighting the Tarrasque, your Eldritch Blast will be entirely useless. Once you're out of spell slots, you're going to be standing around doing nothing. (Slight exaggeration.) Your allies might be able to keep you alive for 10 rounds so that you can get some spells which have a chance of achieving something.
It shouldn't need pointing out, but an hour is 600 rounds, which is just not going to be possible in any combat. (Unless you're fighting a Dire Half-Dragon Fiendish Snail or something, but that's just getting silly.)
Of course, all of this is somewhat pointless if your DM is ok with you doing other things during the minute it takes to use this. For example, even if you're only allowed to use your movement, running around for a minute screaming at your patron for help seems completely achievable. If your DM rules that you can fight as normal during that minute, then this feature might actually be too powerful. Get into a fight with a tough enemy? Start praying to your patron. That gives you 10 rounds to use all your resources, at the end of which you just get them back immediately. Personally, I wouldn't be quite that lenient, but that's up to your DM.
You're correct.
Your analysis is correct--a warlock can cast some long-duration spells by effectively spending hours instead of spell slots.
This is not unbalanced (for a single-classed warlock).
First, the spell slots that the warlock recovers on a short rest can only go up to level 5. Any long-duration spell above level 5 is subject to Mystic Arcanum, which only allows the warlock to cast it once per long rest (as explained here).
So which spells on the list are both level 5 or less and last longer than an hour? My quick perusal through the list shows only Hallucinatory Terrain, Hex, and Dream. Dream is arguably a special case, and Hex only works on one target at a time because of concentration, so your example of Hallucinatory Terrain is essentially the only spell that can be "exploited" this way. Opinions may vary about whether this "exploit" is especially powerful or not, but given that it the spell is quite limited in scope and it requires uninterrupted rests, I'd hardly consider it to be abuse.
Multiclassed warlocks can benefit this way
As Bloodcinder points out, multiclass warlocks can use their warlock spell slots to cast spells from their other classes (PHB 164):
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature...
Thus, a warlock that has levels in wizard could cast a spell like Mage Armor, take a short rest to regain that spell slot, and then enjoy the benefits of Mage Armor for another 7 hours afterward.
As for whether this is balanced, I'd have to assume that the designers considered this possibility because the rule was written out so explicitly. After all, multiclassed spellcasters are delaying (or sacrificing) their ability to cast higher level spells, so there is some tradeoff to be considered.
Best Answer
Yes. You can cast any spell using a slot of the same level as the spell, or higher.
From PHB page 201, Chapter 10: Spellcasting:
and, later on the same page:
Note: In your example, you had both 3rd and 4th level spells slots remaining. You can cast Shield with either a 3rd or 4th level slot but (as there is no enhanced effect for that spell when cast at higher levels - a Shield spell is a Shield spell no matter what level it's cast at), it would be best to cast it as third level spell - keeping the 4th level slot for future 4th-or-lower level spells. Shield doesn't benefit from higher slot levels, but some other level 1-3 spells you might want to cast do.
In other words, cast it with the lowest level slot that will achieve the desired effect, taking into account any possible enhancements from casting at higher slot levels
For example, as mentioned by Mark Cogan, when casting Dispel Magic or Counterspell at higher levels, each increase in slot level increases the level of spell which can be automatically dispelled/interrupted without a spellcasting ability roll.
Similarly, Fireball can be cast using any slot of level 3 or higher - but you don't need to use a 9th level slot against a bunch of kobolds unless you have no lower-level slots remaining.