Rem 700 long action. Mounting a Leupold VX5-HD 3-5X44. Would like to mount it as low as possible with clearance for the bolt handle

My new favorite in rings are the Burris Signature line with the swivelling Nylon(?) inserts, because they avoid the need to lap rings to prevent bending the puny modern scope tubes. They fit into their Universal bases, which, like the Redfield and Leupold ones they ape, have windage adjustments to help mount the scope straight.

If you want to mount the scope central, vertically, to avoid optical atrocity and the hassle of running out of clicks, they also sell eccentric inserts to remove gross misalignment.

Height? I don't know. Medium might work but a 44mm objective could require high. You could google the Burris site or ask gr8fuldoug (who advertises here) or your local gun shop. The problem should not be the bolt handle but the big front bell might touch the barrel or rear sight with medium-height rings.

Medium.
Of any make you prefer. I like warne bases and double lever rings. Mostly for the looks as they never come off anyway.
I use Leupolds too; whatever you like. They all work. And yes, the Burris that Sam likes work well.
If mediums don't work, get high. Then you have a set of mediums for the next project. That is how I do it and I have 20 sets of rings here. That way you never have to ask; you already have them on hand to try.

I also like scopes mounted as low a possible to get a good cheek weld. I have taken a wooden dowel the diameter of the scope tube and wrapped it tightly with tape up to the diameter of the scopes eye piece. You can use any height rings but rings that are too high make the following process easier.

Drop the dowel into the installed lower rings and see if there is bolt clearance. If there is clearance it can be measured and you will know what lower ring would provide clearance. To make this measurement you will need slip "Shim stock" between the bolt and the tape on the dowel. bew playing cards are good for this as you can just add cards to the stack until they just barely slip between the bolt and the tape on the dowel. Make sure you take the "slop" out of the bolt towards the most critical point which checking the clearance. The stack of cards can then be measure to see how much lower rings can be used.

If the rings are not high enough to provide clearance there are couple of things that can be done. You can put a narrow (1/8") piece of wood that is the thickness the extra height of the next higher ring would provide in the center bottom of the lower ring half and center the dowel by putting the upper rings on and using you fingers or loosely installed upper rings screws to maintain the dowel centered over the rings while you check bolt clearance. You can also dispense with the wood and install the top half of the rings and evenly tighten the screws until bolt clearance is lost when the dowel is firmly held up in the upper rings. At this point the gap between the upper rings and the lower rings should be the same and can be measured to determine the minimum extra height that is required. I do my measurement with a digital caliper and if you do not have one they can be cheaply purchased at places like Harbor Freight or from the reloading equipment manufactures.

With the known base and ring heights and the clearance measurements you can also calculate the minimum mounting system height and see which manufacture can provide the closest, but higher; combination of ring and mount height

This can also be done to make sure the ocular will clear the barrel.

I like Talleys and Leupold QD's on most of my guns. I use smaller scopes than most on my hunting rifles, mostly 3X and 4X Leupolds fixed..On my varmint guns I like the fixed 8 the 2x7x28, and the 3x9..Bigger power shows too much mirage in the Idaho high desert..you can get them pretty low with a Redfield, Leupold base and ole WW2 Redfield rings with split rings..Nice set up and I,ve ratholed a number of them, you can also drill and tap them for the LITTLE BO PEEP, A Peep sight that flips up when you remove the scope. I have 3 or 4 of those also.neet. I think you will need a high with a 44 mm objective lens, one reason I use smaller scopes..