Border Gradient With Border Radius

Possible to use border-radius together with a border-image which has a gradient?

Probably not possible, as per the W3C spec:

A box's backgrounds, but not its
border-image, are clipped to the
appropriate curve (as determined by
‘background-clip’). Other effects that
clip to the border or padding edge
(such as ‘overflow’ other than
‘visible’) also must clip to the
curve. The content of replaced
elements is always trimmed to the
content edge curve. Also, the area
outside the curve of the border edge
does not accept mouse events on behalf
of the element.

This is likely because border-image can take some potentially complicated patterns. If you want a rounded, image border, you'll need to create one yourself.

Border Gradient with Border Radius

2021: I recommend using the CSS mask method since the support is pretty good now

You cannot use border-radius with gradient. Here is another idea where you can rely on multiple background and adjust the background-clip:

.white-grad {
  background: 
    linear-gradient(#ccc 0 0) padding-box, /*this is your grey background*/
    linear-gradient(to right, #9c20aa, #fb3570) border-box;
  color: #313149;
  padding: 10px;
  border: 5px solid transparent;
  border-radius: 15px;
  display: inline-block;
  margin: 75px 0;
}

<div class="white-grad"> Some text here</div>

<div class="white-grad"> Some long long long text here</div>

<div class="white-grad"> Some long long <br>long text here</div>

Gradient border with border radius and gradient text

I will consider this previous answer to build the rounded gradient using pseudo element so that you can use background-clip:text on the main element. I have used the mask version by you can also consider the SVG one:

.btn {
  --r:40px; /* radius */
  --b:5px; /* border width */
  
  background: linear-gradient(to right, #006175 0%, #00a950 100%);
  -webkit-background-clip: text;
  background-clip: text;
  -webkit-text-fill-color: transparent;
  color: transparent;
  
  border-radius: var(--r);
  display: flex;
  align-items: center;
  justify-content: center;
  font: 1.5rem 'Oswald', Arial, sans-serif;
  height: 80px;
  margin: 0 auto;
  position: relative;
  z-index:0;
  text-decoration: none;
  width: 264px;
}
/* check lined question for the detail of the below code */
.btn::before {
  content:"";
  position:absolute;
  z-index:-1;
  inset: 0;
  border: var(--b) solid transparent;
  border-radius: var(--r);
  background: inherit;
  background-origin: border-box;
  background-clip: border-box;
  -webkit-mask:
    linear-gradient(#fff 0 0) padding-box,
    linear-gradient(#fff 0 0);
  -webkit-mask-composite: xor;
          mask-composite: exclude;
  -webkit-mask-repeat: no-repeat;
}
/**/
.btn:hover {
  color: #fff;
  -webkit-text-fill-color: #fff;
  -webkit-background-clip: border-box;
  background-clip: border-box;
}

.btn:hover::before {
  -webkit-mask:none;
}

body {
  background:pink;
}

<a class="btn" href="#">
  Click Here!
</a>

border-radius with border-image

It is not possible to combine them. The W3 Spec says:

A box's backgrounds, but not its border-image, are clipped to the appropriate curve (as determined by ‘background-clip’). Other effects that clip to the border or padding edge (such as ‘overflow’ other than ‘visible’) also must clip to the curve. The content of replaced elements is always trimmed to the content edge curve. Also, the area outside the curve of the border edge does not accept mouse events on behalf of the element.

However, you can achieve the same effect by using a css gradient

#cont{  background: -webkit-linear-gradient(left top, crimson 0%, blue 100%);  width: 300px;  height: 300px;  border-radius: 1000px;  padding: 10px;}
#box{  background: white;  width: 300px;  height: 300px;  border-radius: 1000px;}

<div id="cont">  <div id="box"></div></div>

left border gradient fill with border radius

This is not how you will get gradient border, to get a gradient border, you will have to give a background gradient to parent element.Now if you want to get left border as gradient then give padding left to parent element and in similar fashion you can give padding to other sides as well.

I did not touch other parts of your css declaration, just removed the border and added a padding and background color.

 .test{width:300px;
border-radius:20px;
box-shadow:0 0 5px rgba(0,0,0,0.2);
overflow:hidden;
font-family:"Lato",sans-serif;
font-size:16px;
line-height:1;originally 1.5 #9970e0
color:#555555;
margin: 15px;
background: linear-gradient(to bottom,#cc0099,#0099ff);
border-right:0px;
padding: 0 0 0 5px;
}
.inherit{
  width: 100%;
  padding: 0 0 0 1px;
  background: #fff;
}

<div class="test">
<div class="inherit">
Inherit
</div>
</div>