Is STR 3 enough?
To complete the game? Yes. I've done it myself with two completely different builds.
To pass all of the Strength skill checks? No, obviously not.
To wield Sniper Rifles, Assault Rifles, Metal Armor of any kind,
et cetera? No.
For the purposes of an SMG build, you'll be fine. Keep the high DEX and low STR. If the lack of any available inventory space bothers you, then you can consider either taking the Pack Rathound Feat or, and this is what I would recommend, just storing all of your excess junk that you can't sell either in a pile on the ground someplace you'll remember or in your lockers at SGS. I've always personally thought that Pack Rathound is a waste of a Feat, but that's just me. It's your character, after all! Do whatever makes you happiest.
Can I pump all my stat points in DEX?
I mean, you
can, but I don't see why you would need to or really ought to. The max for initial character creation is 10, and the maximum base score is 16 (as you get an extra base ability score every 4 levels, and the max level is 25), but that's not really necessary for a SMG build. Keep the 8 initial DEX, and maybe bump it up to 14 if you want, but that's ultimately up to you. I'd probably spread the extra 6 points evenly between Perception, Dexterity, and Agility if I was playing your build.
Persuasion: maxing necessary?
Not
necessary, per se, but useful. Aim for 100 Persuasion, and then decide whether or not you want to go for the 135 once you hit it. IIRC, the highest persuasion check only requires 120, but I could be wrong or this could have changed.
Throwing/Stealth: pump to 50 base and Electronics/Chemicals as high as I deem necessary for the stuff I want?
Yeah, that sounds smart to me. You can always pump Throwing or Stealth higher if you feel like it's not enough. Conversely, if you hit 30 Throwing and feel that the grenade accuracy you have is good enough, then you can put the extra 20 points towards something else.
For a SMG character, though, you're definitely going to want to prioritize Mechanics over the other crafting disciplines, as SMGs are crafted entirely using Mechanics skill. (Unless, of course, you don't care about upgrading your gun, at which point feel free to allocate your points however you want.)
Keep in mind that, if you're going to be playing stealthily, you're going to want to wear equipment that boosts your Stealth skill in order to help you survive with 50 Stealth come the end-game. Examples of such items: Cloaking Device, Regular Balaclava, Ninja Tabi Boots, and any piece of non-metal armor with a black overcoat.
Do Lockpicking/Hacking need to be maxed?
If you want to unlock and complete as much as you possibly can with your build, then yes, you mind as well max Hacking and Lockpicking by level 25 while you're at it. If you can only afford to max one of the two, then max Hacking because it gets you some neat end-game lore.
Please understand that it is impossible to unlock and discover every little thing in Underrail within the confines of a single build. You can come very, very close, but you just can't do everything. There are going to be skill checks that you cannot pass. Keep this in mind so it doesn't tear you apart inside, if you're that kind of completionist.
Regarding feats, a lot of what is and is not "needed" is subjective. For a SMG build, the Feats that you ought to take will make themselves very obvious when you unlock access to them. As a general rule of thumb, however, your Feat selection (like your Base Ability Scores and Skill Point distribution) should ultimately be a reflection of how you play the game. Make of that what you will.