Check out this question: How can I play monsters and NPCs up to their potential?
The challange and XP ratings assume (I assume) that monsters play to their strengths. For example, your Hobgoblin captain should be commanding the hobgoblins - these are not mindless drones who attack without a plan - these are genetically professional soldiers. Good soldiers do not fight fair - level playing fields are for sporting fixtures, not battles.
The hobgoblins should refuse combat unless and until they have the advantage or they are desperate; but these guys plan ahead so they don't get desperate.
If the hobgoblins are aware of the party's approach then they should be preparing a series of ambushes, fall-back positions, escape/disengagement plans, booby traps, ways of separating the party and destroying them in detail.
If they are unaware then they should have done this anyway - good soldiers do.
For example, an attack on the party during a long rest has possibilities. 7 arrows with surprise at the PC on watch should render them unconscious with a high probability of success. They can then move in and slaughter the remaining PCs. Played this way this is certainly a 2x Deadly encounter. If they survive (which is unlikely) they will have earned every XP several times over.
This assumes that the party really wants "near death battles that bring them out bloody and bruised". My guess is that this is not what they really want.
The suggestion of a "battle arena" test is a good one. This can also be done in the campaign - maybe there are gladiatorial contests in your campaign that are not "to the death". Playing a team knock-out (that ends with 1 team knocked-out) could teach both you and your players what "deadly" can really mean without having a TPK.
Probably the summoned monster doesn't count toward the challenge.
Bear in mind there's no RAW that specifies this, and challenge ratings of published monsters include some degree of "eyeballing" from the writers. But we can extrapolate a bit by using the basic challenge evaluation from the DMG on the stats that we do know.
I used Monstermancer* to run the stats of the Drow Mage without the summoning ability, counting all of its spells, and came up with a baseline challenge of 7:
"eval": {
"defCR": 0.5,
"offCR": 15,
"dmg": 126,
"ac": 15,
"hp": 45,
"atk": 6,
"svDC": 14
}
So, without any eyeballing or special tweaks to its challenge evaluation, the Drow Mage's challenge matches the one given in the Monster Manual (in fact, this might be a little low, since it doesn't factor in the possibility of having an ongoing cloudkill spell active while throwing lightning bolts, but I digress).
However, while it looks like the summoned creature isn't "baked in" to the summoner's challenge rating, there isn't much guidance as to how you should incorporate it into an encounter. It's less effective than a full-fledged combatant for a few reasons (it takes an action to summon, it might not always succeed, &c.), but adding a challenge 4 shadow demon to a fight against a single challenge 7 mage is not a minor impact on the encounter, so it should count for something.
For a little more circumstantial evidence, there are variants listed in the Monster Manual for demons and devils which can summon other demons and devils; there is an entry for each type of fiend and which others they can summon with a percentage chance, similar to the ability of the drow mage cited here. These variants don't mention a change to the fiends' challenge ratings. Since the book doesn't assume a difference in the creature's own challenge with or without the summon ability there, it follows that the summoned creature should count separately in some way (even if not as a full-fledged combatant).
*Caveat: Monstermancer isn't perfect at this stuff, but it's an OK starting point. Also, there's a huge mismatch between the offensive and defensive challenges it gives to the Drow Mage, so it's a lot easier to take out than its challenge rating implies.
Best Answer
The important point to remember here is that this decision only affects the multiplier on the encounter to determine how difficult it is. Therefore the points to note are:
Therefore it is primarily a judgement call that does not have huge implications. To illustrate consider the examples that you have given:
CR3 Mummy plus (3) CR 1/2 Swarms of Beetles
XP = 700 + 3 x 100 = 1000
Multiplier: 1 or 2 giving 1000 or 2000
CR3 Goblin plus CR1 Lieutenant plus (5) CR 1/4 Swarms of Beetles
XP = 700 + 200 + 5 x 50 = 1050
Multiplier: 1 or 1.5 or 2.5 giving 1050 or 1575 or 2625.
Against say 4 Level 3 PCs, in both cases the first multiplier rates this as "Hard" and any of the others are "Deadly" - what this says is that these are dangerous encounters; be careful. The low CR creatures will have a significant impact on the fight. So using them in the multiplier is reasonable because they low CR creatures will use up significantly more of the party's resources.
Against say 4 level 6 PCs, they are "Easy" to "Medium"; but the low CR creatures will probably be rendered hors de combat in Round 1 - their only contribution is to distract the PCs for that round and maybe burn a fireball. Not using the multiplier would be reasonable here because said fireball will probably be damaging the main protagonist as well so there is no additional drain on party resources.
CR10 and CR1
XP = 5900 + 200 = 6100
Multiplier: 1 or 1.5 giving 6100 or 9150.
This is a deadly encounter for the parties above with the first guy alone (and a probable TPK for the L3 guys).
For 4 level 10 PCs its "Medium" without the multiplier and "Hard" with it. However, given that the CR1 monster can probably be dropped with a Cantrip (3d10), Single attack or at most a 1st level spell (Sleep) it doesn't really add to the difficulty - he will distract 1 of the 4 PCs for part of 1 round. Ignore it.