Can you make a weapon attack with a weapon you cast telekinesis upon?
Yes. Moving a thing in any way with telekinesis, fine or otherwise, consumes an action. This means you may use your action to do any of your regular combat actions, (such as prepare, dash, attack, cast a spell, etc.) Or you can use your action to manipulate your targeted object. It effectively gives you an extra action type. The spell puts no restrictions on your fine movement of an object, so there is no reason for why you couldn't swing a sword or mace around. It might not be possible to fire a crossbow with any accuracy though, and you wouldn't be able to use any other ranged weapon either, as all of those require you to interact with the weapon and its ammunition at the same time, while telekinesis is strictly limited to one target object.
If they don't say no, the answer is yes. The game's rules and content represent realistic things, so they are capable of doing anything the real equivalent would be capable of. The DMG provides DMs a wide variety of abstract tools which allow them to represent the infinite properties of a simulated reality. As such, as long as the rules do not specifically exclude something, and the DM believes it is within an accurate representation, then the thing really does have the assumed properties. This includes everything from humans having hands and feet, gravity pulling objects downward, polearms having a wooden shaft that can interact with the environment the same way a 10ft pole would, and all sorts of other elements of the game. D&D depends on the assumption of representation in order to function at all.
Since it consumes the same action resource as an attack, the only benefit it gives you is field control, as you are now effectively occupying two locations on the map at the same time, and you may make melee attacks without putting yourself in harm's way of reciprocal melee attacks on subsequent turns. As a level 5 spell, this seems fairly reasonable, especially compared to the significant damage output possible from other 5th level spells.
Even in a general sense, simply passing an object through their space would effectively be like attacking them. You could, for example, lift a boulder several times larger than them and drag it through their space, effectively smearing them across the floor like a giant boot to a bug, you could drop it on them for an identical effect. You could even squash them against a wall with it like a bulldozer. The same effect could be achieved by running a lance or spear through their space, essentially the same as stabbing them while riding by on horseback. You could even do things like strangle a person with their own necklace. (Since you aren't removing it from their person, the spell vs. strength contest is not invoked)
As for how the "attack" would be resolved, that would depend on how your DM interprets the nature of magic, spells, damage, and attacks. In other words, it would come down to what the DM thinks the rules represent.
Stupid Magic: The spell is primarily kinesthetic/somatic in nature. Moving the target object would be the same as moving your own limbs. In this regard, damage could probably be resolved in the same fashion as an attack roll for simplicity.
Smart Magic: The spell does the execution for you. This would be more like thinking "attack that guy" and the spell carrying out your desire/command. This interpretation would be more likely to use your spell attack modifier or some such.
Environmental Effect: This interpretation would treat the "attack" as nothing more than a collision. In this case, the DM would determine damage based on the rules for improvising damage in the DMG.
Would that make it a spell attack?
No. A spell attack is magical in its own right. In this case, it's more like attacking a person while wearing magic gloves. The magic is touching the weapon, but that doesn't make the weapon magical, and the telekinesis isn't directly causing harm to the target.