No, you can't use spiritual weapon as a melee weapon
This is a really cool idea but unfortunately doesn't work by RAW or RAI.
Spells only do what they say and no part of spiritual weapon indicates that you can use it as a melee weapon.
Spiritual Weapon states:
You create a floating weapon that lasts for the duration or until you cast this spell again.
So I can certainly see where this idea came from. However it specifies you make a melee spell attack as a bonus action. So that is the only attack it can make.
As a bonus action, you can move the weapon up to 20 feet and repeat the attack against a creature within 5 feet of it.
The only way to move spiritual weapon is to use your bonus action to move it 20ft. There is no way to gain additional bonus actions in a single turn so you can't move it further than this.
WARNING THIS IS UNTESTED. I am not recommending this but providing it as a base if you wish to homebrew this into your game.
The second part of your question has potential for homebrew.
Could I even just grab it and run thirty feet, then release it so it can move attack for my bonus action?
Using your action this way seems like a balanced option to me. Allowing movement shouldn't be too game-breaking but would be definitely not supported by RAW.
There may be some unintended consequences to allowing this. As Rubiksmoose points out, the weapon is not a physical object at all and cannot be grabbed. If you can grab it so can other creatures. This completely changes the nature of the spell. Does it now occupy a space? Can it be damaged? etc.
As spiritual weapon can take any form you wish, allowing it to be a physical object would bring it fairly close to the 14th level illusion schools Illusory Reality feature. Though not as powerful this is still extremely strong for a 1st level spell and you would want to limit how it could be used.
The first question allowing you to use your action to attack with spiritual weapon is broken and should not be allowed. Cast at first level this may not seem like a big deal. 1d8 + Spellcasting Modifier is equal or less than most melee weapons you would normally be wielding.
The problem is when casting at higher levels, at 5th level spiritual weapon deals 3d8 force damage. Allowing attacks with action and bonus action would mean 6d8 force damage per turn, potentially up to 60d8 over the duration for a single 5th level spell slot.
Allowing it to be used as a weapon also exposes you to the question of making attacks of opportunity with it. Spiritual Weapon cannot make attacks of opportunity by RAW and allowing this would only exacerbate the problem with this house rule.