Not Heavy Explosives, no.
The floor of UFOs (including the outside roof, which counts as a floor, as far as the game is concerned) are definitely destructible, but are quite durable. The internal floors / walls are slightly less durable.
As you've seen yourself, shooting down a UFO causes one (or more) of their engines to explode, which has a chance to destroy the roof.
Unfortunately, terrain destruction requires a certain threshold of damage to trigger, and the only item in X-Com's arsenal strong enough to breach the hull of a UFO is the Blaster Launcher.
It is worth noting that explosives always do a fixed amount of damage to terrain, so if one HE explosive didn't blow it up (and/or change the tile to an alternate, "Damaged" tile), the second one won't either. Terrain can't take partial damage, so you can't detonate several HE's over a couple of turns to get around this, either.
Finally, Explosives in cannot stack. It is not viable, for instance, to throw 3 HE onto the same tile to increase the damage dealt; the terrain will only take damage from the first.
This page in the UFOPaedia has more.
Improving your armor should probably be a priority, improved armor will prevent grenades from killing half your soldiers in one turn, but you'll still get severe injuries, and blaster bombs will still usually kill those on the center nine tiles of the blast.
Best options
There are two ways that you can prevent losing anyone to a grenade: kill (or mind control) all of the aliens close enough to throw one (or better yet, all aliens that can see you, to prevent blaster bombs as well), or use flying armor to keep your soldiers off the ground, so they can't be hit by grenades.
Mind control is probably your best option late game: if you can scout one alien, mind controlling it will often reveal several more, allowing you to quickly get the vision you need to eliminate or take control of all nearby enemies, denying them vision of your units (especially important when going up against psionic aliens or blaster bombs).
Flying, once you have it, is a good choice for your back-row snipers that tend to get clustered tightly on the first few turns, but I find flying scouts tend to be more easily spotted by aliens and often fall to plasma fire, so use it in moderation.
Early game options
The alternatives early on are not quite as flexible: you can minimize the probability of being surprised if you get vision in all directions on turn one, you can attempt to spread your soldiers out more so fewer can be caught in a blast, or you can bring out fewer soldiers at once if your first scouts don't see nearby aliens that need killing, to minimize damage if your scouting failed.
You should be able to prevent this from becoming something that happens every time, but you can't prevent it entirely. Improved armor will prevent it from killing those not at the center of the blast, but you'll still get severe injuries.
Coping with darkness
If it's a night mission, then getting proper vision to eliminate the aliens nearby is more difficult, since the aliens' vision is not reduced by low light, but yours is, making losses to grenades much more likely. Starting your soldiers near the door with a flare in their hand and having them toss it around 18 tiles away can give you full vision on the first turn, however (both soldiers and flares provide a nine-tile radius of light high enough to see aliens, not counting the center tile, so 19 tiles is as far as you should consider throwing it, but probably a few less to prevent blind spots is better).
If the terrain is flammable (e.g. jungle), I'm also a fan of giving a solider with good strength but abysmal aim an auto-cannon, and having them fire an auto shot of incendiary ammo, starting several fires spread widely across the battlefield (the worse the soldier's aim, the more widely the fires will be spread, so this is not as effective with a good soldier). Long distance snap shots are also a good option for lighting up the battlefield before scouting further.
Best Answer
Yes it is!
However, early game you're unlikely to have appropriate
satelliteRadar coverage to detect the Terror ships, and with only avalanche missiles, you'll either need a lot of interceptors, or a great deal of luck (especially considering the attack range bug), since they can destroy interceptors in two shots, assuming you can even catch up to the ships in time. They are one of the two Large-class UFOs (the other being the Supply Ship).Once you've got your detection network a little better off, you may have better odds. But until you've got Hyperwave Decoder and some better guns, your chance of detecting, interecpting, and then downing, a terrorship are remote.
You can also take "preventative" measures. Terror Missions are scouted (with Scout UFOS to start with) before the final run on a city is made. If you're consistent about shooting down these precursors, you can often delay or avoid the terror mission entirely. As before, though, hitting the right UFOs is a question of luck, before you get the Hyperwave Decoder, and the Aliens eventually use stronger ships to scout, so you still should upgrade your fighter weapons ASAP.