I’ll try to address each part of your question individually

What would a tank division/regiment that still had a good number of tanks, but suffered significant casualties to it's mechanized/motorized infantry do? Keep pushing forward? Pull back and wait for replacements?….Stay in place and take a more defensive role?

Alongside infantry divisions, armored divisions suffered the most from the ups and downs of both the Zone of Interior and European Theater replacement systems. They were able to consistently maintain a high numerical personnel strength relative to their allowed table of organization even after horrendous casualties, but this was often accomplished via the receipt of a varied percentage of men with little or no infantry training that had to be indoctrinated on the spot, or even after entering combat. These untrained or poorly trained men, as well as men trained as infantry, but that had never experienced combat before, often caused heavy losses among veteran officers and noncommissioned officers who had to go "out in front" and show them what to do and how to do it.

Vehicles were another problem, in that an untrained or poorly trained vehicle meant no vehicle. There was a consistent shortage of medium tanks in relation to the T/O&E strength of armored units in the European theater because of bureaucratic failures from June 1944 until the spring of 1945. The War Department had initially set the European theater replacement rate for medium tanks at 7 percent of total T/O&E strength per month, basing it on factors from North Africa and Italy. This rate was raised to 9 percent just before D-Day. Losses in June and July (around 25 percent per month) were such that by August 1944, no reserve of medium tanks existed, and by September, the First Army was forced to adopt provisional tables of organization which reduced the number of medium tanks in its armored divisions and separate tank battalions. The replacement rate was raised to 11 percent in October (based on a combined loss rate of 9.9 percent in Europe and Italy in July 1944), and to 14 percent in December. Actual losses were higher, although the War Department, the European theater, and the field armies often frustratingly clashed when reporting this issue between themselves. The European Theater used a rate of 13.1 percent during its appeal to an Army Service Forces mission for immediate shipments in December 1944 (see below), the 12th Army Group offered a cumulative rate of 20 percent through October 1944, and the field armies repeatedly claimed an exaggerated figure of a 25 percent cumulative loss rate up to December 1944.

The situation saw no improvement during the fall months. By the end of November there were on hand in the theater only 3,344 tanks against a T/O&E requirement of 3,409 and an authorized on-hand reserve requirement of 937. No true reserve existed, therefore. The 12th Army Group reported that two of its tank battalions had fewer than ten serviceable tanks, and field commanders in general deplored the fact that armored units had to operate at from 10 to 25 percent below authorized strength. Furthermore, the current theater troop basis failed to provide the one tank battalion per infantry division which field commanders considered a necessary minimum. As a partial remedy steps were taken to convert two battalions of the 10th Armored Group to composite tank battalions for use with infantry divisions. The field commands had continued to urge the theater to obtain a higher replacement factor, arguing that a larger flow of replacement tanks was imperative if the habitual infantry-tank co-operation which had characterized all operations thus far was to continue. The 12th Army Group noted that at no time since the middle of August had the armies had their full T/O&E allowance of tanks, and that not since the early days of the Normandy beachhead had they possessed a reserve. It maintained that a 25 percent reserve in each army was an operational necessity.

In December 1944, "the month of reckoning for several other major logistic problems, partly as the result of the first-hand investigation which Lt. Gen. LeRoy Lutes, Director of Operations, ASF, made during his visit to the theater," the European Theater successfully persuaded the War Department to release 1,010 additional medium tanks for emergency shipment in January 1945, but the Battle of the Bulge upset all estimates. The situation became so critical that 351 tanks were immediately "reverse Lend-Leased" from the British (who maintained a sizable reserve at all times). 75 tanks mistakenly delivered to Marseille but intended for delivery to the Mediterranean, plus an additional 75 in Italy, were diverted from the Mediterranean Theater. It was later decided to "assign the entire output of U.S. production to U.S. forces until their reserves in Europe totaled 2,000, which was expected to require four months."

Even understrength, armored divisions continued to push on.

The fighting strength of the 4th Armored Division had drained away perceptibly as its tanks were destroyed and its veteran infantry and tank crews were reduced in action after action.^20 But in spite of the fact that it had been fighting under conditions ill suited for armored action, the division had succeeded in fulfilling the mission of armor and had rolled up the German defense line on the Sarre for a distance of twenty miles across the front of the XII Corps zone of advance....

20.) Losses in experienced officer personnel had been very high since 8 November: a number of companies had had a 100 percent turnover in officers, and the battalion staffs had been decimated. The armored infantry, of course, suffered high casualties. On 2 December the 51st Armored Infantry Battalion had a rifle strength of 160 and the 53d had only 126. Company B, 51st Armored Infantry Battalion, received 128 replacements for losses between 9 November and 6 December. Tank losses also were severe. The 8th Tank Battalion, already crippled by the loss of nearly a whole medium tank company at Baerendorf, lost 14 tanks in operations between 3 and 6 December. XII Corps G-3 Jnl, 2 Dec 44. The casualties inflicted on the tank crews were particularly damaging to the division since the replacements coming in at the end of the Lorraine Campaign usually were converted riflemen or headquarters personnel who knew nothing about tanks. In one batch of replacements sent to the 37th Tank Battalion only two men had ever seen the inside of a tank--one of these as the result of having once been given a ride by a cousin in the Armored Force while training in the United States. During the latter part of 1944, training at the Armored Replacement Training Center had deteriorated and few replacements were being turned out. See The Armored Force Command and Center, AGF Study No. 27 (Historical Division, War Department).

Battle fatigue rates in the armored divisions were by far the highest in the armored infantry units, for a variety of reasons, and armored infantry units suffered some of the highest battle fatigue rates of any single ground arm.

The symptoms of BF [battle fatigue] vary. A soldier may become increasingly emotional -- crying easily, becoming irritable, using excessive profanity. Or he may experience sleep disturbances (have difficulty going to sleep or have nightmares), and exaggerated responses to sudden nearby noises and movements....

Units with high BF rates either had poor leadership or their leaders had become battle or psychiatric casualties. Within the infantry in World War II, armored infantry units sustained the highest rate of BF casualties followed by regular infantry units, and then by such volunteer units as the airborne infantry and the Rangers. (Interestingly enough, although these volunteer infantry units had the fewest BF casualties, they sustained significantly higher battle casualty rates.)

The high BF rates among armored infantrymen can be attributed to several factors. These infantrymen resented their relative lack of mobile protection (trucks and half-tracks as compared to armored tanks); they were belittled by other infantrymen and the tankers; they had to put up with more primitive living conditions than the tankers they accompanied; and the tactical situation often required them to conduct dismounted (and unprotected) operations during the most hazardous times....

A “formerly secret” U.S. Army medical report states the following. I have been unable to track down the source due to Osprey Publications’ unwillingness to allow a full bibliography in their shorter titles.

In armored divisions, with too few armored infantry, the infantry contributes from 80 to 90% of the combat exhaustion casualties, rates becoming extremely high after the third to fifth days of action. In hard, continued action, armored infantry companies may be down to 40–50 men (out of a strength of 245), with [all] three company commanders casualties in the process. One unit had 150-180% replacements in 200 days; another had 100% turnover in 60-70 days.

For operations, the lack of infantry in armored divisions was often compensated for by temporarily attaching normal infantry units (generally a battalion or regiment) from nearby infantry divisions. "Motorized" infantry divisions which had enough trucks to move themselves in one go and could keep pace with armored divisions were tried briefly in 1942 and 1943. The 4th, 6th, 7th, 8th, and 90th Infantry Divisions were redesignated motorized divisions, but the concept was rejected in March 1943 and four were converted back. The 4th Motorized Division was to use halftracks instead of trucks for its infantry, and remained motorized until August 1943 pending potential overseas use, when it was officially rejected by commanders due to the the amount of rubber used in its vehicles (318 tons versus 166 tons in a normal infantry division) and demands on shipping space.