Yes, you can cast Flame Blade while holding a shield. In general, you need one free hand to cast a spell. Any material and somatic components required can be performed jointly with that one hand. There's nothing in the description of flame blade that requires a second free hand, so this rule is not superseded. Thus, you can hold a shield in your left hand, cast flame blade with your right hand, receive the blade in your right hand, and happily immolate your foes as you please.
As for concentration, unless something specifically says otherwise, it is only broken by the caster's conscious decision, by taking damage, or by casting another concentration spell.
Flame blade has a duration of concentration, up to 10 minutes, while flaming sphere has a duration of concentration, up to 1 minute and shillelagh a duration of one minute (no concentration).
Flame blade provides 3d6 magical fire damage using your Wisdom modifier (melee spell attack) for 10 minutes. This could potentially span two or three combats (likely, if you're in a tightly packed dungeon), and allows you to use other spells or cantrips as you please as long as they don't require concentration.
Flaming sphere, on the other hand, typically lasts no longer than one combat (10 rounds at most), and uses the same level spell slot as flame blade.
We can do the math and see that economically, when the hit ratio is the same (as you have pointed out in your question), flaming sphere does only 1d6 avg "guaranteed" damage, and can only do that 10 times. I say "guaranteed" because it only does damage when it rams an enemy or when an enemy stops within 5 feet of it, and the enemy still gets a save against it. You can control when it rams an enemy, you can't control when an enemy stops near it.
On the other hand, flame blade can hit for 3d6 (that's an additional 4 average damage over flaming sphere if the target fails its save), and it can do it 10 times as many times. At an average of 11 damage and assuming a perfect hit ratio and attacking every round, that's 1100 potential damage over 10 minutes. Flaming sphere, on the other hand, can do 70 at most (and under perfect conditions, with everything going just right, 560 total avg damage).
Finally, flame blade has utility beyond being a weapon: it also produces light that can be dismissed at will and evoked using a bonus action while the spell is still within its casting duration. This is more useful than the light provided by flaming sphere, which in this case burns twice as bright but only a tenth as long.
Best Answer
You "make a spell attack" as described on PHB p.205: you roll a d20 and add your proficiency bonus and spellcasting modifier. (Given that you're probably a druid, that'll be d20+PROF+WIS.)
Damage is then 3d6 + nothing, or more if you cast above 2nd-level.