You can do whatever you like; what I would do is play it as written.

PHB p. 144

If the Armor table shows “Str 13” or “Str 15” in the Strength column for an armor type, the armor reduces the wearer’s speed by 10 feet unless the wearer has a Strength score equal to or higher than the listed score.

As it says, if they are strong enough they suffer no penalty to speed, if not then they lose 10 feet of speed.

PHB p. 182

Climbing , Swimming , and Crawling

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. ... Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.

As it says, a creature without a swim speed uses 2 feet of movement for every 1 foot covered (3 if difficult terrain). A creature with a swim speed moves 1 for 1 (2 for difficult terrain). As phrased, a swim speed does not exempt you from making a Strength (Athletics) check if required.

Significantly, it says nothing about the effects of armor on swimming.

Does this make a lot of sense? From a game perspective, yes - it gives simple rules that can be used with a minimum of complexity.

Can you rule that armor should make a difference? Of course, it says you can right on p. 6 of the PHB - if the PC says "I swim across in my Plate armor" then a perfectly sensible ruling is "You sink like a stone", however, given that this is not what the rules say, fairness dictates that you tell the player the consequence of their action before they undertake it and also before they make a commitment to spend in game money on an expensive anchor.

DMG p. 159

CLOAK OF THE MANTA RAY

Wondrous item, uncommon

While wearing this cloak with its hood up, you can breathe underwater, and you have a swimming speed of 60 feet. Pulling the hood up or down requires an action.

As it says, it gives you a swim speed - take that and apply it to the rules above.

TL;DR

Question 1

"can they swim 60 ft as the DMG says (pg 159)"?

Yes

Question 2

"are any of you DMs ruling that there are some penalties to swimming in heavy armor even with a swim speed?"

I cannot speak for "any DM", but by the rules in the book: There are no penalties to swimming in armor with or without a swim speed.

Question 3

"Would you include a higher Str requirement with swimming?"

Me? No, by the rules in the book: The cloak gives you a swim speed of 60 feet. If you meet the strength requirement, there is no penalty to your speed; if you do not then your speed is reduced by 10 feet (i.e. to 50 feet).

There are magic and non-magic items made from adamantine

While the DMG lists Adamantine Armor as a magic item, some items made from adamantine are not magic.

Gargoyles resist damage from “nonmagical weapons that aren’t adamantine” (MM, p 140). It follows there would be nonmagical weapons that are adamantine.

So, adamantine weapons would strike gargoyles similarly to how magic weapons would, without being magic weapons, or even magic items. Mechanically, they are similar to silvered weapons (the phrasing in monsters’ resistances blocks are the same) but they defeat a different set of monsters’ damage resistances.

Xanathar’s Guide does not call Adamantium “Magical”

Xanathar’s Guide to Everything (p. 78) has a section on adamantium weapons which describes adamatine as “an ultrahard metal found in meteorites and extraordinary mineral veins” but does not mention it being magical.

Xanathar’s lists the properties of weapons “made from or coated by” adamantine. These are distinct from the proprties of magic items. For example, adamantine weapons don’t get magic item resiliency.

Official Example (Spoiler Alert)

There is an example of such a weapon in official Wizard’s 5e material.

In The Lost Mine of Phandelver, the Spider Staff is a “black, adamantine staff” that “can be wielded as a quarterstaff.”

Note, the staff does not strike as a “magic quarterstaff” like the Staff of Power does. It is a magic item but not a magic weapon. As far as whomping things goes, it is a “nonmagical weapon made of adamantine.”

For mithral items, the rules say very little

I’m not aware of any mention of mithral in the core rules outside of the magic armor. A DM might follow the example of adamantine — that it’s a rare metal that is often found in magic items. Or they might rule it is inherently magical.

Magic Items are defined as such. Custom items are defined by your DM.

If an item is listed as a magic item then in general they would have the qualities common to magic items of their type. A custom item invented by your DM might have any properties.

The particular qualities of any particular item in your game are of course up your DM. Your character might (and probably should) learn whether an item is magical or not, and its properties, resistances, etc., by means such as an Identify spell.